Section 5.1 Adding and Subtracting Polynomials
¶Objectives: PCC Course Content and Outcome Guide
A polynomial is a particular type of algebraic expression.
A company's sales, \(s\) (in millions of dollars), can be modeled by \(2.2t+5.8\text{,}\) where \(t\) stands for the number of years since \(2010\text{.}\)
The height of an object from the ground, \(h\) (in feet), launched upward from the top of a building can be modeled by \(-16t^2+32t+300\text{,}\) where \(t\) represents the amount of time (in seconds) since the launch.
The volume of an open-top box with a square base, \(V\) (in cubic inches), can be calculated by \(30s^2-\frac{1}{2}s^2\text{,}\) where \(s\) stands for the length of the square base, and the box sides have to be cut from a certain square piece of metal.
All of the expressions above are polynomials. In this section, we will learn some basic vocabulary relating to polynomials and we'll then learn how to add and subtract polynomials.
Subsection 5.1.1 Polynomial Vocabulary
There is a lot of vocabulary associated with polynomials. We start this section with a flood of vocabulary terms and some examples of how to use them.
Definition 5.1.2.
A polynomial is an expression with one or more terms summed together. A term of a polynomial must either be a plain number or the product of a number and one or more variables raised to natural number powers. The expression \(0\) is also considered a polynomial, with zero terms.
Example 5.1.3.
Here are three polynomials: \(x^2-5x+2\text{,}\) \(t^3-1\text{,}\) \(7y\text{.}\)
The expression \(3x^4y^3+7xy^2-12xy\) is an example of a polynomial in more than one variable.
The polynomial \(x^2-5x+3\) has three terms: \(x^2\text{,}\) \(-5x\text{,}\) and \(3\text{.}\)
The polynomial \(3x^4+7xy^2-12xy\) also has three terms.
The polynomial \(t^3-1\) has two terms.
Remark 5.1.4.
A polynomial will never have a variable in the denominator of a fraction or under a radical.
Definition 5.1.5.
The coefficient (or numerical coefficient) of a term in a polynomial is the numerical factor in the term.
Example 5.1.6.
The coefficient of the term \(\frac{4}{3}x^6\) is \(\frac{4}{3}\text{.}\)
The coefficient of the second term of the polynomial \(x^2-5x+3\) is \(-5\text{.}\)
The coefficient of the term \(\frac{y^7}{4}\) is \(\frac{1}{4}\text{.}\)
Checkpoint 5.1.7.
Definition 5.1.8.
A term in a polynomial with no variable factor is called a constant term.
Example 5.1.9.
The constant term of the polynomial \(x^2-5x+3\) is \(3\text{.}\)
Definition 5.1.10.
The degree of a term is one way to measure how âlargeâ it is. When a term only has one variable, its degree is the exponent on that variable. When a term has more than one variable, its degree is the sum of the exponents on the variables. A constant term has degree \(0\text{.}\)
Example 5.1.11.
The degree of \(5x^2\) is \(2\text{.}\)
The degree of \(-\frac{4}{7}y^5\) is \(5\text{.}\)
The degree of \(-4x^2y^3\) is \(5\text{.}\)
The degree of \(17\) is \(0\text{.}\) Constant terms always have \(0\) degree.
Definition 5.1.12.
The degree of a nonzero polynomial is the greatest degree that appears amongst its terms.
Definition 5.1.13.
The leading term of a polynomial is the term with the greatest degree (assuming there is no tie). The coefficient of a polynomial's leading term is called the polynomial's leading coefficient.
Example 5.1.14.
The degree of the polynomial \(4x^2-5x+3\) is \(2\) because the terms have degrees \(2\text{,}\) \(1\text{,}\) and \(0\text{,}\) respectively, and \(2\) is the largest. Its leading term is \(4x^2\text{,}\) and its leading coefficient is \(4\text{.}\)
Remark 5.1.15.
To help us recognize a polynomial's degree, the standard convention at this level is to write a polynomial's terms in order from highest degree to lowest degree. When a polynomial is written in this order, it is written in standard form. For example, it is standard practice to write \(7-4x-x^2\) as \(-x^2-4x+7\) since \(-x^2\) is the leading term. By writing the polynomial in standard form, we can look at the first term to determine both the polynomial's degree and leading term.
There are special names for polynomials with a small number of terms, and for polynomials with certain degrees.
- monomial
A polynomial with one term, such as \(3x^5\text{,}\) is called a monomial.
- binomial
A polynomial with two terms, such as \(3x^5+2x\text{,}\) is called a binomial.
- trinomial
A polynomial with three terms, such as \(x^2-5x+3\text{,}\) is called a trinomial.
- constant polynomial
A zeroth-degree polynomial is called a constant polynomial. An example is the polynomial \(7\text{,}\) which has degree zero.
- linear polynomial
A first-degree polynomial is called a linear polynomial. An example is \(-2x+7\text{.}\)
- quadratic polynomial
A second-degree polynomial is called a quadratic polynomial. An example is \(4x^2-2x+7\text{.}\)
- cubic polynomial
A third-degree polynomial is called a cubic polynomial. An example is \(x^3+4x^2-2x+7\text{.}\)
Fourth-degree and fifth-degree polynomials are called quartic and quintic polynomials, respectively. If the degree of the polynomial, \(n\text{,}\) is greater than five, we'll simply call it an \(n\)th-degree polynomial. For example, the polynomial \(5x^8-4x^5+1\) is an \(8\)th-degree polynomial.
Subsection 5.1.2 Adding and Subtracting Polynomials
Example 5.1.16. Production Costs.
Bayani started a company that makes one product: one-gallon ketchup jugs for industrial kitchens. The company's production expenses only come from two things: supplies and labor. The cost of supplies, \(S\) (in thousands of dollars), can be modeled by \(S=0.05x^2+2x+30\text{,}\) where \(x\) is number of thousands of jugs of ketchup produced. The labor cost for his employees, \(L\) (in thousands of dollars), can be modeled by \(0.1x^2+4x\text{,}\) where \(x\) again represents the number of jugs they produce (in thousands of jugs). Find a model for the company's total production costs.
Since Bayani's company only has these two costs, we can find a model for the total production costs, \(C\) (in thousands of dollars), by adding the supply costs and the labor costs:
To finish simplifying our total production cost model, we'll combine the like terms:
This simplified model can now calculate Bayani's total production costs \(C\) (in thousands of dollars) when the company produces \(x\) thousand jugs of ketchup.
In short, the process of adding two or more polynomials involves recognizing and then combining the like terms.
Checkpoint 5.1.17.
Example 5.1.18.
Simplify the expression \(\left(\frac{1}{2}x^2-\frac{2}{3}x-\frac{3}{2}\right)+\left(\frac{3}{2}x^2+\frac{7}{2}x-\frac{1}{4}\right)\text{.}\)
Example 5.1.19. Profit, Revenue, and Costs.
From Example 5.1.16, we know Bayani's ketchup company's production costs, \(C\) (in thousands of dollars), for producing \(x\) thousand jugs of ketchup is modeled by \(C=0.15x^2+6x+30\text{.}\) The revenue, \(R\) (in thousands of dollars), from selling the ketchup can be modeled by \(R=13x\text{,}\) where \(x\) stands for the number of thousands of jugs of ketchup sold. The company's net profit can be calculated using the concept:
Assuming all products produced will be sold, a polynomial to model the company's net profit, \(P\) (in thousands of dollars) is:
The key distinction between the addition and subtraction of polynomials is that when we subtract a polynomial, we must subtract each term in that polynomial.
Notice that our first step in simplifying the expression in Example 5.1.19 was to subtract every term in the second expression. We can also think of this as distributing a factor of \(-1\) across the second polynomial, \(0.15x^2+6x+30\text{,}\) and then adding these terms as follows:
Example 5.1.20.
Subtract \(\left(5x^3+4x^2-6x\right)-\left(-3x^2+9x-2\right)\text{.}\)
We must first subtract every term in \(\left(-3x^2+9x-2\right)\) from \(\left(5x^3+4x^2-6x\right)\text{.}\) Then we can combine like terms.
Checkpoint 5.1.21.
Let's look at one last example where the polynomial has multiple variables. Remember that like terms must have the same variable(s) with the same exponent.
Example 5.1.22.
Subtract \(\left( 3x^2y+8xy^2-17y^3 \right)-\left(2x^2y+11xy^2+4y^2 \right)\text{.}\)
Again, we'll begin by subtracting each term in \(\left(2x^2y+11xy^2+4y^2\right)\text{.}\) Once we've done this, we'll need to identify and combine like terms.
Subsection 5.1.3 Evaluating Polynomial Expressions
Evaluating expressions was introduced in Section 1.1, and involves replacing the variable(s) in an expression with specific numbers and calculating the result. Here, we will look at evaluating polynomial expressions.
Example 5.1.23.
Evaluate the expression \(-12y^3+4y^2-9y+2\) for \(y=-5\text{.}\)
We will replace \(y\) with \(-5\) and simplify the result:
Recall that in Subsection 1.1.4 and Example 1.1.15 we discussed how \((-5)^2\) and \(-5^2\) are not the same expressions. The first expression, \((-5)^2\text{,}\) represents the number \(-5\) squared, and is \((-5)(-5)=25\text{.}\) The second expression, \(-5^2\text{,}\) is the opposite of the number that you get after you square \(5\text{,}\) and is \(-5^2=-(5\cdot 5) = -25\text{.}\)
Example 5.1.24.
Evaluate the expression \(C=0.15x^2+6x+30\) from Example 5.1.16 for \(x=10\) and explain what this means in context.
We will replace \(x\) with \(10\text{:}\)
The context was that \(x\) represents so many thousands of jugs of ketchup, and \(C\) represents the total cost, in thousands of dollars, to produce that many jugs. So in context, we can interpret this as it costing \(\$105{,}000\) to produce \(10{,}000\) jugs of ketchup.
Checkpoint 5.1.25.
Reading Questions 5.1.4 Reading Questions
1.
What are the names for a polynomial with one term? With two terms? With three terms? Care to take a guess at the name of a polynomial with four terms?
2.
Adding and subtracting polynomials is mostly about combining terms.
3.
What should you be careful with when evaluating a polynomial for a negative number?
Exercises 5.1.5 Exercises
Review and Warmup
1.
List the terms in each expression.
\({6s-0.9z+3.5}\)
\({-y-6.5z^{2}}\)
\({1.6y^{2}+6.7z-0.2x-2.2t^{2}}\)
\({-0.5x+6.2z-2+3.4y^{2}}\)
2.
List the terms in each expression.
\({7.6s^{2}+7.3s^{2}+1.5}\)
\({-2.2z^{2}}\)
\({-4.1x-2.5t^{2}}\)
\({-3.9t+8.1z^{2}+6.7z^{2}+7.4x}\)
3.
List the terms in each expression.
\({-8.9s-2.6-0.6x+2.8y}\)
\({-2.6t-6.8x^{2}+6.4}\)
\({5.8z^{2}+4.7s^{2}+3.4}\)
\({-8.4y^{2}}\)
4.
List the terms in each expression.
\({-7.3t^{2}+5.6t-2.5-4.7z^{2}}\)
\({5.9y^{2}+3.1t^{2}}\)
\({-8.5s^{2}}\)
\({-6.8t^{2}+4.7t^{2}+6.4+3.6s^{2}}\)
5.
Simplify each expression, if possible, by combining like terms.
\({-6t-5t-5t-6t}\)
\({2t-3t+8s+4t}\)
\({9s^{2}+2s^{2}+7z^{2}}\)
\({-4y^{2}+7y^{2}-5z}\)
6.
Simplify each expression, if possible, by combining like terms.
\({-4t^{2}+4y^{2}}\)
\({-6s+3+6s^{2}}\)
\({5s^{2}-4t+3s+6}\)
\({7z^{2}+5y^{2}-6x^{2}}\)
7.
Simplify each expression, if possible, by combining like terms.
\({-{\frac{3}{8}}t - {\frac{1}{9}}t}\)
\({-{\frac{2}{3}}x+7s-2s}\)
\({-{\frac{7}{8}}z - {\frac{5}{7}}z+{\frac{7}{8}}z}\)
\({-{\frac{6}{7}}s^{2} - {\frac{1}{5}}t^{2}}\)
8.
Simplify each expression, if possible, by combining like terms.
\({{\frac{1}{3}}t^{2} - {\frac{2}{3}}y}\)
\({-{\frac{2}{9}}s^{2}+{\frac{8}{9}} - {\frac{1}{9}}z^{2}+2}\)
\({-{\frac{6}{5}}z^{2} - {\frac{3}{8}}s^{2}-s^{2}}\)
\({{\frac{3}{5}}s^{2}+{\frac{1}{2}}y - {\frac{1}{8}}s^{2}+{\frac{1}{3}}}\)
Vocabulary Questions
9.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{4y^{12}-13y^{3}}}\) is a
monomial
binomial
trinomial
10.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{-11r^{7}+4r^{2}}}\) is a
monomial
binomial
trinomial
11.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{40}}\) is a
monomial
binomial
trinomial
12.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{5}}\) is a
monomial
binomial
trinomial
13.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{-18y^{11}-9y^{7}-20y^{6}}}\) is a
monomial
binomial
trinomial
14.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{-20r^{10}-2r^{9}-10r^{2}}}\) is a
monomial
binomial
trinomial
15.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{8x^{3}+17x^{7}+6x}}\) is a
monomial
binomial
trinomial
16.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{-14x^{7}-16x^{8}+6x}}\) is a
monomial
binomial
trinomial
17.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{13y^{11}}}\) is a
monomial
binomial
trinomial
18.
Is the following expression a monomial, binomial, or trinomial?
\(\displaystyle{{-2y^{19}}}\) is a
monomial
binomial
trinomial
19.
Find the degree of the following polynomial.
\(\displaystyle{ {2x^{8}y^{6}-16x^{2}y^{4}-6x^{2}+13} }\)
20.
Find the degree of the following polynomial.
\(\displaystyle{ {6x^{7}y^{9}+11x^{3}y+11x^{2}+1} }\)
Simplifying Polynomials
21.
Add the polynomials.
\(\displaystyle{\left({6x-2}\right)+\left({-7x-5}\right)}\)
22.
Add the polynomials.
\(\displaystyle{\left({8x-9}\right)+\left({2x+10}\right)}\)
23.
Add the polynomials.
\(\displaystyle{\left({10x^{2}+5x}\right)+\left({-10x^{2}+4x}\right)}\)
24.
Add the polynomials.
\(\displaystyle{\left({-8x^{2}-3x}\right)+\left({-x^{2}-2x}\right)}\)
25.
Add the polynomials.
\(\displaystyle{\left({-3x^{2}-9x+1}\right)+\left({4x^{2}-7x+3}\right)}\)
26.
Add the polynomials.
\(\displaystyle{\left({4x^{2}+6x-9}\right)+\left({7x^{2}+8x+2}\right)}\)
27.
Add the polynomials.
\(\displaystyle{\left({4y^{3}-7y^{2}-4}\right)+\left({-3y^{3}-4y^{2}-7}\right)}\)
28.
Add the polynomials.
\(\displaystyle{\left({-10r^{3}-4r^{2}+6}\right)+\left({-4r^{3}+7r^{2}+1}\right)}\)
29.
Add the polynomials.
\(\displaystyle{\left({7r^{6}-2r^{4}-4r^{2}}\right)+\left({4r^{6}-10r^{4}-r^{2}}\right)}\)
30.
Add the polynomials.
\(\displaystyle{\left({4t^{6}-8t^{4}+6t^{2}}\right)+\left({-5t^{6}+3t^{4}-9t^{2}}\right)}\)
31.
Add the polynomials.
\(\displaystyle{\left({0.8t^{5}-0.3t^{4}-0.1t^{2}-0.4}\right)+\left({0.4t^{5}-0.9t^{3}+0.4}\right)}\)
32.
Add the polynomials.
\(\displaystyle{\left({0.2t^{5}+0.5t^{4}-0.6t^{2}-0.1}\right)+\left({0.5t^{5}+0.2t^{3}-0.8}\right)}\)
33.
Add the polynomials.
\(\displaystyle{\left({-2x^{3}+3x^{2}-5x+{\frac{7}{6}}}\right)+\left({3x^{3}-10x^{2}+3x+{\frac{5}{4}}}\right)}\)
34.
Add the polynomials.
\(\displaystyle{\left({3x^{3}+7x^{2}+5x+{\frac{7}{10}}}\right)+\left({-5x^{3}-7x^{2}+2x+{\frac{1}{4}}}\right)}\)
35.
Subtract the polynomials.
\(\displaystyle{\left({-4x+1}\right)-\left({-10x-3}\right)}\)
36.
Subtract the polynomials.
\(\displaystyle{\left({-x-6}\right)-\left({-x-8}\right)}\)
37.
Subtract the polynomials.
\(\displaystyle{\left({x^{2}+7x}\right)-\left({7x^{2}-3x}\right)}\)
38.
Subtract the polynomials.
\(\displaystyle{\left({3x^{2}-5x}\right)-\left({x^{2}-3x}\right)}\)
39.
Subtract the polynomials.
\(\displaystyle{\left({-5x^{9}-3x^{4}}\right)-\left({-5x^{3}-6}\right)}\)
40.
Subtract the polynomials.
\(\displaystyle{\left({10x^{10}-4x^{8}}\right)-\left({6x-4}\right)}\)
41.
Subtract the polynomials.
\(\displaystyle{\left({-10x^{2}+6x-4}\right)-\left({-4x^{2}+8x+2}\right)}\)
42.
Subtract the polynomials.
\(\displaystyle{\left({-2x^{2}-10x-4}\right)-\left({-8x^{2}+3x-10}\right)}\)
43.
Subtract the polynomials.
\(\displaystyle{\left({-8x^{6}-3x^{4}-5x^{2}}\right)-\left({7x^{6}-3x^{4}-5x^{2}}\right)}\)
44.
Subtract the polynomials.
\(\displaystyle{\left({5y^{6}-9y^{4}+5y^{2}}\right)-\left({7y^{6}+6y^{4}+8y^{2}}\right)}\)
45.
Subtract the polynomials.
\(\displaystyle{\left({-5x^{3}+3x^{2}-5x-5}\right)-\left({-8x^{2}-6x+7}\right)}\)
46.
Subtract the polynomials.
\(\displaystyle{\left({6x^{3}-7x^{2}-5x+5}\right)-\left({9x^{2}+2x-7}\right)}\)
47.
Add or subtract the given polynomials as indicated.
\(\displaystyle{ \left[{4r^{16}-10r^{15}+r^{14}} - \left({-8r^{16}+3r^{15}-2r^{14}}\right)\right] - \left({-9r^{16}-7r^{15}-8r^{14}}\right) }\)
48.
Add or subtract the given polynomials as indicated.
\(\displaystyle{ \left[{t^{9}+8t^{8}} - \left({-8t^{9}-10t^{8}}\right)\right] - \left({-5t^{9}-4t^{8}}\right) }\)
49.
Add or subtract the given polynomials as indicated.
\(\displaystyle{ \left[{7t^{13}+5t^{12}} - \left({-9t^{13}-10t^{12}}\right)\right] - \left[{-10t^{13}+2t^{12}}+\left({-10t^{13}-7t^{12}}\right)\right] }\)
50.
Add or subtract the given polynomials as indicated.
\(\displaystyle{ \left[{4t^{14}-8t^{13}+3t^{5}} - \left({-9t^{14}+8t^{13}-9t^{5}}\right)\right] - \left[{-7t^{14}-8t^{13}+7t^{5}}+\left({-4t^{14}-10t^{13}-8t^{5}}\right)\right] }\)
51.
Add or subtract the given polynomials as indicated.
\(\displaystyle{ \left({2x^{7}y^{3}+8xy}\right)+\left({-3x^{7}y^{3}+4xy}\right) }\)
52.
Add or subtract the given polynomials as indicated.
\(\displaystyle{ \left({9x^{4}y^{3}-10xy}\right)+\left({2x^{4}y^{3}+9xy}\right) }\)
53.
Add or subtract the given polynomials as indicated.
\(\displaystyle{ \left({10x^{9}y^{7}+6xy+8}\right)+\left({9x^{9}y^{7}+9xy+9}\right) }\)
54.
Add or subtract the given polynomials as indicated.
\(\displaystyle{ \left({5x^{8}y^{3}-10xy-6}\right)+\left({3x^{8}y^{3}-6xy+3}\right) }\)
55.
Add or subtract the given polynomials as indicated.
\(\left({6x^{8}y^{9}+5x^{5}y^{3}+9xy}\right)+\left({-3x^{8}y^{9}-7x^{5}y^{3}+8xy}\right)\)
56.
Add or subtract the given polynomials as indicated.
\(\left({-7x^{7}y^{8}+8x^{3}y^{4}+6xy}\right)+\left({3x^{7}y^{8}-2x^{3}y^{4}+7xy}\right)\)
57.
Add or subtract the given polynomials as indicated.
\(\left({8x^{6}-3xy+5y^{9}}\right)-\left({-2x^{6}-9xy+2y^{9}}\right)\)
58.
Add or subtract the given polynomials as indicated.
\(\left({9x^{5}+7xy-8y^{6}}\right)-\left({2x^{5}+10xy-5y^{6}}\right)\)
59.
Add or subtract the given polynomials as indicated.
\(\left({-10x^{7}y^{6}+2x^{2}y^{4}+2xy}\right)-\left({-2x^{7}y^{6}-10x^{2}y^{4}+5xy}\right)\)
60.
Add or subtract the given polynomials as indicated.
\(\left({2x^{8}y^{9}-6x^{3}y^{4}-3xy}\right)-\left({-4x^{8}y^{9}+10x^{3}y^{4}-8xy}\right)\)
61.
Add or subtract the given polynomials as indicated.
\(\left({3x^{4}-9y^{2}}\right)-\left({-2x^{4}+3x^{2}y^{2}+7x^{4}y^{2}-2y^{2}}\right)\)
62.
Add or subtract the given polynomials as indicated.
\(\left({-4x^{9}+4y^{4}}\right)-\left({2x^{9}-10x^{8}y^{4}+4x^{9}y^{4}-10y^{4}}\right)\)
63.
Subtract \(-4y^{18}-8y^{7}-6y^{5}\) from the sum of \(9y^{18}-4y^{7}+7y^{5}\) and \(-10y^{18}+3y^{7}-7y^{5}\text{.}\)
64.
Subtract \(-10r^{11}-6r^{5}-2r^{3}\) from the sum of \(5r^{11}-9r^{5}+4r^{3}\) and \(-10r^{11}+7r^{5}-7r^{3}\text{.}\)
65.
Subtract \({-7x^{3}y^{7}-6xy}\) from \({-2x^{3}y^{7}-9xy}\)
66.
Subtract \({8x^{3}y^{8}+10xy}\) from \({10x^{3}y^{8}+7xy}\)
Evaluating Polynomials
67.
Evaluate the expression \({t^{2}}\text{:}\)
For \(t=5\text{.}\)
For \(t=-4\text{.}\)
68.
Evaluate the expression \({t^{2}}\text{:}\)
For \(t=2\text{.}\)
For \(t=-8\text{.}\)
69.
Evaluate the expression \({-x^{2}}\text{:}\)
For \(x=4\text{.}\)
For \(x=-2\text{.}\)
70.
Evaluate the expression \({-x^{2}}\text{:}\)
For \(x=3\text{.}\)
For \(x=-4\text{.}\)
71.
Evaluate the expression \({y^{3}}\text{:}\)
For \(y=2\text{.}\)
For \(y=-2\text{.}\)
72.
Evaluate the expression \({y^{3}}\text{:}\)
For \(y=4\text{.}\)
For \(y=-3\text{.}\)
73.
Evaluate \({\left(-2r\right)^{2}}\) when \(r=-1\text{.}\)
Evaluate \({\left(-2r\right)^{3}}\) when \(r=-1\text{.}\)
74.
Evaluate \({\left(-r\right)^{2}}\) when \(r=-3\text{.}\)
Evaluate \({\left(-r\right)^{3}}\) when \(r=-3\text{.}\)
75.
Evaluate the expression \(\displaystyle \frac{1}{3} \big( x + 3 \big)^2 - 7\) when \(x = -6\text{.}\)
76.
Evaluate the expression \(\displaystyle \frac{1}{5} \big( x + 4 \big)^2 - 4\) when \(x = -9\text{.}\)
77.
Evaluate the expression \(\displaystyle \frac{1}{3} \big( x + 4 \big)^2 - 9\) when \(x = -7\text{.}\)
78.
Evaluate the expression \(\displaystyle \frac{1}{4} \big( x + 1 \big)^2 - 6\) when \(x = -5\text{.}\)
79.
Evaluate the expression \(-16t^{2}+64t+128\) when \(t=-3\text{.}\)
80.
Evaluate the expression \(-16t^{2}+64t+128\) when \(t=-5\text{.}\)
81.
Evaluate the expression \(-16t^{2}+64t+128\) when \(t=-4\text{.}\)
82.
Evaluate the expression \(-16t^{2}+64t+128\) when \(t=2\text{.}\)
Applications of Simplifying Polynomials
83.
The formula
gives the vertical position of an object, at time \(t\text{,}\) thrown with an initial velocity \(v_0\text{,}\) from an initial position \(y_0\) in a place where the acceleration of gravity is \(a\text{.}\) The acceleration of gravity on earth is \({-9.8\ {\textstyle\frac{\rm\mathstrut m}{\rm\mathstrut s^{2}}}}\text{.}\) It is negative, because we consider the upward direction as positive in this situation, and gravity pulls down.
What is the height of a baseball thrown with an initial velocity of \(v_0={82\ {\textstyle\frac{\rm\mathstrut m}{\rm\mathstrut s}}}\text{,}\) from an initial position of \(y_0= {94\ {\rm m}}\text{,}\) and at time \(t={1\ {\rm s}}\text{?}\)
One second after the baseball was thrown, it was high in the air.
84.
The formula
gives the vertical position of an object, at time \(t\text{,}\) thrown with an initial velocity \(v_0\text{,}\) from an initial position \(y_0\) in a place where the acceleration of gravity is \(a\text{.}\) The acceleration of gravity on earth is \({-9.8\ {\textstyle\frac{\rm\mathstrut m}{\rm\mathstrut s^{2}}}}\text{.}\) It is negative, because we consider the upward direction as positive in this situation, and gravity pulls down.
What is the height of a baseball thrown with an initial velocity of \(v_0={87\ {\textstyle\frac{\rm\mathstrut m}{\rm\mathstrut s}}}\text{,}\) from an initial position of \(y_0= {76\ {\rm m}}\text{,}\) and at time \(t={9\ {\rm s}}\text{?}\)
Nine seconds after the baseball was thrown, it was high in the air.
85.
An auto companyâs sales volume (in dollars) can be modeled by \({6.1x^{2}+6.9x+4}\text{,}\) and its costs (in dollars) can be modeled by \({4.6x^{2}+3.7x+4}\text{,}\) where \(x\) represents the number of cars produced. We can calculate the companyâs net profit by subtracting cost from sales. Find the polynomial which models the companyâs profit in thousands of dollars.
The companyâs profit can be modeled by dollars.
86.
An auto companyâs sales volume (in dollars) can be modeled by \({8.4x^{2}+1.2x+3.9}\text{,}\) and its costs (in dollars) can be modeled by \({4.9x^{2}-2.2x+3.9}\text{,}\) where \(x\) represents the number of cars produced. We can calculate the companyâs net profit by subtracting cost from sales. Find the polynomial which models the companyâs profit in thousands of dollars.
The companyâs profit can be modeled by dollars.
87.
A farmhand is building two pig pens of the same width, but different lengths. If \(x\) is the common width, in meters, then the cost of building one pen is \({34x^{2}-4x+22}\) dollars, and the cost of building the other pen is \({22x^{2}+4x+46.5}\) dollars. Whatâs the total cost of building those two pens?
A polynomial representing the total cost of building those two pens is dollars.
88.
A farmhand is building two pig pens of the same width, but different lengths. If \(x\) is the common width, in meters, then the cost of building one pen is \({23.5x^{2}+8x+20.5}\) dollars, and the cost of building the other pen is \({25.5x^{2}-8x+19}\) dollars. Whatâs the total cost of building those two pens?
A polynomial representing the total cost of building those two pens is dollars.
89.
A farmer is building fence around a triangular area. The cost of building the shortest side is \({35x}\) dollars, where \(x\) stands for the length of the side in feet. The cost of building the other two sides can be modeled by \({8x^{2}+0.5x+40}\) dollars and \({4x^{3}-4x+35}\) dollars, respectively. Whatâs the total cost of building fence for all three sides?
The cost of building fence for all three sides would be dollars.
90.
A farmer is building fence around a triangular area. The cost of building the shortest side is \({40x}\) dollars, where \(x\) stands for the length of the side in feet. The cost of building the other two sides can be modeled by \({5x^{2}+4.5x+30}\) dollars and \({4x^{3}+1.5x+35}\) dollars, respectively. Whatâs the total cost of building fence for all three sides?
The cost of building fence for all three sides would be dollars.
91.
An architect is designing a house on an empty plot. The area of the plot can be modeled by the polynomial \({4x^{4}+6x^{2}-2.5x}\text{,}\) and the area of the houseâs base can be modeled by \({6x^{3}-2.5x+15}\text{.}\) The rest of the plot is the yard. Whatâs the yardâs area?
The area of the yard can be modeled by the polynomial .
92.
An architect is designing a house on an empty plot. The area of the plot can be modeled by the polynomial \({5x^{4}+16x^{2}+6x}\text{,}\) and the area of the houseâs base can be modeled by \({4x^{3}+6x+15}\text{.}\) The rest of the plot is the yard. Whatâs the yardâs area?
The area of the yard can be modeled by the polynomial .