6.1 Add and Subtract Polynomials

Before you get started, take this readiness quiz.

Simplify: $8 x + 3 x .$
If you missed this problem, review the reference.

Subtract: $\left(5 n + 8\right) - \left(2 n - 1\right) .$
If you missed this problem, review the reference.

Write in expanded form: $a^{5} .$
If you missed this problem, review the reference.

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial.

1. ⓐ $4 y^{2} - 8 y - 6$
2. ⓑ $−5 a^{4} b^{2}$
3. ⓒ $2 x^{5} - 5 x^{3} - 9 x^{2} + 3 x + 4$
4. ⓓ $13 - 5 m^{3}$
5. ⓔ $q$

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

ⓐ $5 b$ ⓑ $8 y^{3} - 7 y^{2} - y - 3$ ⓒ $−3 x^{2} - 5 x + 9$ ⓓ $81 - 4 a^{2}$ ⓔ $−5 x^{6}$

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

ⓐ $27 z^{3} - 8$ ⓑ $12 m^{3} - 5 m^{2} - 2 m$ ⓒ $\frac{5}{6}$ ⓓ $8 x^{4} - 7 x^{2} - 6 x - 5$ ⓔ $− n^{4}$

Find the degree of the following polynomials.

1. ⓐ $10 y$
2. ⓑ $4 x^{3} - 7 x + 5$
3. ⓒ $−15$
4. ⓓ $−8 b^{2} + 9 b - 2$
5. ⓔ $8 x y^{2} + 2 y$

Find the degree of the following polynomials:

ⓐ $−15 b$ ⓑ $10 z^{4} + 4 z^{2} - 5$ ⓒ $12 c^{5} d^{4} + 9 c^{3} d^{9} - 7$ ⓓ $3 x^{2} y - 4 x$ ⓔ $−9$

Find the degree of the following polynomials:

ⓐ $52$ ⓑ $a^{4} b - 17 a^{4}$ ⓒ $5 x + 6 y + 2 z$ ⓓ $3 x^{2} - 5 x + 7$ ⓔ $− a^{3}$

Add: $25 y^{2} + 15 y^{2}$.

Add: $12 q^{2} + 9 q^{2} .$

Add: $−15 c^{2} + 8 c^{2} .$

Subtract: $16 p - \left(−7 p\right)$.

Subtract: $8 m - \left(−5 m\right) .$

Subtract: $−15 z^{3} - \left(−5 z^{3}\right) .$

Simplify: $c^{2} + 7 d^{2} - 6 c^{2}$.

Add: $8 y^{2} + 3 z^{2} - 3 y^{2} .$

Add: $3 m^{2} + n^{2} - 7 m^{2} .$

Simplify: $u^{2} v + 5 u^{2} - 3 v^{2}$.

Simplify: $m^{2} n^{2} - 8 m^{2} + 4 n^{2} .$

Simplify: $p q^{2} - 6 p - 5 q^{2} .$

Find the sum: $\left(7 x^{2} - 4 x + 5\right) + \left(x^{2} - 7 x + 3\right) .$

Find the sum: $\left(14 y^{2} + 6 y - 4\right) + \left(3 y^{2} + 8 y + 5\right) .$

Find the difference: $\left(8 x^{2} + 3 x - 19\right) - \left(7 x^{2} - 14\right) .$

Find the difference: $\left(9 b^{2} - 5 b - 4\right) - \left(3 b^{2} - 5 b - 7\right) .$

Subtract: $\left(5 z^{2} - 6 z - 2\right)$ from $\left(7 z^{2} + 6 z - 4\right)$.

Subtract: $\left(x^{2} - 5 x - 8\right)$ from $\left(6 x^{2} + 9 x - 1\right)$.

Find the sum: $\left(u^{2} - 6 u v + 5 v^{2}\right) + \left(3 u^{2} + 2 u v\right)$.

Find the sum: $\left(3 x^{2} - 4 x y + 5 y^{2}\right) + \left(2 x^{2} - x y\right)$.

Find the sum: $\left(2 x^{2} - 3 x y - 2 y^{2}\right) + \left(5 x^{2} - 3 x y\right)$.

Find the difference: $\left(p^{2} + q^{2}\right) - \left(p^{2} + 10 p q - 2 q^{2}\right)$.

Find the difference: $\left(a^{2} + b^{2}\right) - \left(a^{2} + 5 a b - 6 b^{2}\right)$.

Find the difference: $\left(m^{2} + n^{2}\right) - \left(m^{2} - 7 m n - 3 n^{2}\right)$.

Simplify: $\left(a^{3} - a^{2} b\right) - \left(a b^{2} + b^{3}\right) + \left(a^{2} b + a b^{2}\right)$.

Simplify: $\left(x^{3} - x^{2} y\right) - \left(x y^{2} + y^{3}\right) + \left(x^{2} y + x y^{2}\right)$.

Simplify: $\left(p^{3} - p^{2} q\right) + \left(p q^{2} + q^{3}\right) - \left(p^{2} q + p q^{2}\right)$.

Evaluate: $3 x^{2} + 2 x - 15$ when

1. ⓐ $x = 3$
2. ⓑ $x = −5$
3. ⓒ $x = 0$

Evaluate: $5 z^{2} - z - 4$ when

1. ⓐ $z = −2$
2. ⓑ $z = 0$
3. ⓒ $z = 2$

The polynomial $−16 t^{2} + 250$ gives the height of a ball $t$ seconds after it is dropped from a 250 foot tall building. Find the height after $t = 2$ seconds.

The polynomial $−16 t^{2} + 250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall building. Find the height after $t = 0$ seconds.

The polynomial $−16 t^{2} + 250$ gives the height of a ball $t$ seconds after it is dropped from a 250-foot tall building. Find the height after $t = 3$ seconds.

The polynomial $6 x^{2} + 15 x y$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with $x = 6$ feet and $y = 4$ feet.

The polynomial $6 x^{2} + 15 x y$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with $x = 5$ feet and $y = 8$ feet.

Determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

ⓐ $81 b^{5} - 24 b^{3} + 1$ ⓑ $5 c^{3} + 11 c^{2} - c - 8$ ⓒ $\frac{14}{15} y + \frac{1}{7}$
ⓓ 5
ⓔ $4 y + 17$

Determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

ⓐ $8 - 3 x$ ⓑ $z^{2} - 5 z - 6$ ⓒ $y^{3} - 8 y^{2} + 2 y - 16$ ⓓ $81 b^{5} - 24 b^{3} + 1$ ⓔ $−18$

Determine the degree of each polynomial.

ⓐ $6 a^{2} + 12 a + 14$ ⓑ $18 x y^{2} z$ ⓒ $5 x + 2$ ⓓ $y^{3} - 8 y^{2} + 2 y - 16$ ⓔ $−24$

Determine the degree of each polynomial.

ⓐ $14 - 29 x$ ⓑ $z^{2} - 5 z - 6$ ⓒ $y^{3} - 8 y^{2} + 2 y - 16$ ⓓ $23 a b^{2} - 14$ ⓔ $−3$

Add or subtract the monomials.

$\left( \text{7x}\right)^{2} + 5 x^{2}$

Add or subtract the monomials.

$−12 w + 18 w$

Add or subtract the monomials.

$\text{4a} - 9 a$

Add or subtract the monomials.

$28 x - \left(−12 x\right)$

Add or subtract the monomials.

$−5 b - 17 b$

Add or subtract the monomials.

$12 a + 5 b - 22 a$

Add or subtract the monomials.

$2 a^{2} + b^{2} - 6 a^{2}$

Add or subtract the monomials.

$x y^{2} - 5 x - 5 y^{2}$

Add or subtract the monomials.

$a^{2} b - 4 a - 5 a b^{2}$

Add or subtract the monomials.

$\text{12a} + 8 b$

Add or subtract the monomials.

Add: $4 a , −3 b , −8 a$

Add or subtract the monomials.

Subtract $5 x^{6} \text{from} - 12 x^{6}$.

Add or subtract the polynomials.

$\left(5 y^{2} + 12 y + 4\right) + \left(6 y^{2} - 8 y + 7\right)$

Add or subtract the polynomials.

$\left(x^{2} + 6 x + 8\right) + \left(−4 x^{2} + 11 x - 9\right)$

Add or subtract the polynomials.

$\left(8 x^{2} - 5 x + 2\right) + \left(3 x^{2} + 3\right)$

Add or subtract the polynomials.

$\left(5 a^{2} + 8\right) + \left(a^{2} - 4 a - 9\right)$

Add or subtract the polynomials.

$\left(4 m^{2} - 6 m - 3\right) - \left(2 m^{2} + m - 7\right)$

Add or subtract the polynomials.

$\left(a^{2} + 8 a + 5\right) - \left(a^{2} - 3 a + 2\right)$

Add or subtract the polynomials.

$\left(12 s^{2} - 15 s\right) - \left(s - 9\right)$

Add or subtract the polynomials.

Subtract $\left(9 x^{2} + 2\right)$ from $\left(12 x^{2} - x + 6\right)$.

Add or subtract the polynomials.

Subtract $\left(7 w^{2} - 4 w + 2\right)$ from $\left(8 w^{2} - w + 6\right)$.

Add or subtract the polynomials.

Find the sum of $\left(2 p^{3} - 8\right)$ and $\left(p^{2} + 9 p + 18\right)$.

Add or subtract the polynomials.

Find the sum of $\left(8 a^{3} - 8 a\right)$ and $\left(a^{2} + 6 a + 12\right)$.

Add or subtract the polynomials.

Find the difference of
$\left(w^{2} + w - 42\right)$ and
$\left(w^{2} - 10 w + 24\right)$.

Add or subtract the polynomials.

Find the difference of
$\left(c^{2} + 4 c - 33\right)$ and
$\left(c^{2} - 8 c + 12\right)$.

Add or subtract the polynomials.

$\left(7 x^{2} - 2 x y + 6 y^{2}\right) + \left(3 x^{2} - 5 x y\right)$

Add or subtract the polynomials.

$\left(7 m^{2} + m n - 8 n^{2}\right) + \left(3 m^{2} + 2 m n\right)$

Add or subtract the polynomials.

$\left(a^{2} - b^{2}\right) - \left(a^{2} + 3 a b - 4 b^{2}\right)$

Add or subtract the polynomials.

$\left(u^{2} - v^{2}\right) - \left(u^{2} - 4 u v - 3 v^{2}\right)$

Add or subtract the polynomials.

$\left(p^{3} - 3 p^{2} q\right) + \left(2 p q^{2} + 4 q^{3}\right) - \left(3 p^{2} q + p q^{2}\right)$

Add or subtract the polynomials.

$\left(x^{3} - x^{2} y\right) - \left(4 x y^{2} - y^{3}\right) + \left(3 x^{2} y - x y^{2}\right)$

Evaluate each polynomial for the given value.

Evaluate $8 y^{2} - 3 y + 2$ when:

ⓐ $y = 5$ ⓑ $y = −2$ ⓒ $y = 0$

Evaluate each polynomial for the given value.

Evaluate $4 - 36 x$ when:

ⓐ $x = 3$ ⓑ $x = 0$ ⓒ $x = −1$

Evaluate each polynomial for the given value.

A painter drops a brush from a platform 75 feet high. The polynomial $−16 t^{2} + 75$ gives the height of the brush $t$ seconds after it was dropped. Find the height after $t = 2$ seconds.

Evaluate each polynomial for the given value.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $−4 p^{2} + 420 p .$ Find the revenue received when $p = 60$ dollars.

Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of $x$ miles per hour is given by the polynomial $- \frac{1}{150} x^{2} + \frac{1}{3} x$. Find the fuel efficiency when $x = 30 \text{mph}$.

Rental Cost The cost to rent a rug cleaner for $d$ days is given by the polynomial $5.50 d + 25$. Find the cost to rent the cleaner for 6 days.

Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial $\frac{9}{5} c + 32$ where $c$ represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when $c = 65 ° .$

Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

Jonathan thinks that $\frac{1}{3}$ and $\frac{1}{x}$ are both monomials. What is wrong with his reasoning?