5.1 Add and Subtract Polynomials

Before you get started, take this readiness quiz.

Simplify: $3 x^{2} + 3 x + 1 + 8 x^{2} + 5 x + 5 .$
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.

Evaluate: $4 x y^{2}$ when $x = −2$ and $y = 5 .$
If you missed this problem, review the reference.

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ $7 y^{2} - 5 y + 3$ ⓑ $−2 a^{4} b^{2}$ ⓒ $3 x^{5} - 4 x^{3} - 6 x^{2} + x - 8$ ⓓ $2 y - 8 x y^{3}$ ⓔ 15

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

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

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial. Then, find the degree of each polynomial.

ⓐ $64 k^{3} - 8$ ⓑ $9 m^{3} + 4 m^{2} - 2$ ⓒ $\frac{5}{6}$ ⓓ $8 a^{4} - 7 a^{3} b - 6 a^{2} b^{2} - 4 a b^{3} + 7 b^{4}$ ⓔ $− p^{4} q^{3}$

Add or subtract: ⓐ $25 y^{2} + 15 y^{2}$ ⓑ $16 p q^{3} - \left(−7 p q^{3}\right) .$

Add or subtract: ⓐ $12 q^{2} + 9 q^{2}$ ⓑ $8 m n^{3} - \left(−5 m n^{3}\right) .$

Add or subtract: ⓐ $−15 c^{2} + 8 c^{2}$ ⓑ $−15 y^{2} z^{3} - \left(−5 y^{2} z^{3}\right) .$

Simplify: ⓐ $a^{2} + 7 b^{2} - 6 a^{2}$ ⓑ $u^{2} v + 5 u^{2} - 3 v^{2} .$

Add: ⓐ $8 y^{2} + 3 z^{2} - 3 y^{2}$ ⓑ $m^{2} n^{2} - 8 m^{2} + 4 n^{2} .$

Add: ⓐ $3 m^{2} + n^{2} - 7 m^{2}$ ⓑ $p q^{2} - 6 p - 5 q^{2} .$

Find the sum:$\left(7 y^{2} - 2 y + 9\right) + \left(4 y^{2} - 8 y - 7\right) .$

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: $\left(9 w^{2} - 7 w + 5\right) - \left(2 w^{2} - 4\right) .$

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 $\left(p^{2} + 10 p q - 2 q^{2}\right)$ from $\left(p^{2} + q^{2}\right) .$

Subtract $\left(a^{2} + 5 a b - 6 b^{2}\right)$ from $\left(a^{2} + b^{2}\right) .$

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

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

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

For the function $f \left(x\right) = 3 x^{2} + 2 x - 15 ,$ find ⓐ $f \left(3\right)$ ⓑ $f \left(−5\right)$ ⓒ $f \left(0\right) .$

For the function $g \left(x\right) = 5 x^{2} - x - 4 ,$ find ⓐ $g \left(−2\right)$ ⓑ $g \left(−1\right)$ ⓒ $g \left(0\right) .$

The polynomial function $h \left(t\right) = −16 t^{2} + 250$ 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 function $h \left(t\right) = −16 t^{2} + 150$ gives the height of a stone t seconds after it is dropped from a 150-foot tall cliff. Find the height after $t = 0$ seconds (the initial height of the object).

The polynomial function $h \left(t\right) = −16 t^{2} + 175$ gives the height of a ball t seconds after it is dropped from a 175-foot tall bridge. Find the height after $t = 3$ seconds.

For functions $f \left(x\right) = 2 x^{2} - 4 x + 3$ and $g \left(x\right) = x^{2} - 2 x - 6 ,$ find: ⓐ $\left(f + g\right) \left(x\right)$ ⓑ $\left(f + g\right) \left(3\right)$ ⓒ $\left(f - g\right) \left(x\right)$ ⓓ $\left(f - g\right) \left(−2\right) .$

For functions $f \left(x\right) = 5 x^{2} - 4 x - 1$ and $g \left(x\right) = x^{2} + 3 x + 8 ,$ find ⓐ $\left(f + g\right) \left(x\right)$ ⓑ $\left(f + g\right) \left(3\right)$ ⓒ $\left(f - g\right) \left(x\right)$ ⓓ $\left(f - g\right) \left(−2\right) .$

Determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

ⓐ $47 x^{5} - 17 x^{2} y^{3} + y^{2}$
ⓑ $5 c^{3} + 11 c^{2} - c - 8$
ⓒ $\frac{5}{9} a b + \frac{1}{3} b$
ⓓ 4
ⓔ $4 p q + 17$

Determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

ⓐ $8 y - 5 x$
ⓑ $y^{2} - 5 y z - 6 z^{2}$
ⓒ $y^{3} - 8 y^{2} + 2 y - 16$
ⓓ $81 a b^{4} - 24 a^{2} b^{2} + 3 b$
ⓔ $−18$

Determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

ⓐ $5 a^{2} + 12 a b - 7 b^{2}$
ⓑ $18 x y^{2} z$
ⓒ $5 x + 2$
ⓓ $y^{3} - 8 y^{2} + 2 y - 16$
ⓔ $−24$

Determine if the polynomial is a monomial, binomial, trinomial, or other polynomial. Then, indicate the degree of the polynomial.

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

Add or subtract the monomials.

ⓐ $\left( \text{7x}\right)^{\text{2}} + 5 x^{2}$
ⓑ $\text{4a} - 9 a$

Add or subtract the monomials.

ⓐ $−12 w + 18 w$
ⓑ $7 x^{2} y - \left(−12 x^{2} y\right)$

Add or subtract the monomials.

$\left( \text{7x}\right)^{\text{2}} + 5 x^{2} + \text{4a} - 9 a$

Add or subtract the monomials.

$−12 w + 18 w + 7 x^{2} y - \left(−12 x^{2} y\right)$

Add or subtract the monomials.

ⓐ $−5 b - 17 b$
ⓑ $3 x y - \left(−8 x y\right) + 5 x y$

Add or subtract the monomials.

ⓐ $\text{12} a + 5 b - 22 a$
ⓑ $p q^{2} - 4 p - 3 q^{2}$

Add or subtract the monomials.

ⓐ $2 a^{2} + b^{2} - 6 a^{2}$
ⓑ $x^{2} y - 3 x + 7 x y^{2}$

Add or subtract the monomials.

ⓐ $x y^{2} - 5 x - 5 y^{2}$
ⓑ $\text{19y} + 5 z$

Add or subtract the monomials.

$\text{14x} - 3 y - 13 x + a^{2} b - 4 a - 5 a b^{2}$

Add or subtract the monomials.

$5 u^{2} + 4 v^{2} - 6 u^{2} + \text{12a} + 8 b$

Add or subtract the monomials.

Add: $4 a , −3 b , −8 a$

Add or subtract the monomials.

Subtract $5 x^{6}$ from $−12 x^{6}$

Add the polynomials.

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

Add the polynomials.

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

Add the polynomials.

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

Add the polynomials.

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

Subtract the polynomials.

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

Subtract the polynomials.

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

Subtract the polynomials.

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

Subtract the polynomials.

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

Subtract the polynomials.

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

Find the difference of the polynomials.

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

Add the polynomials.

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

Add 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.

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

Add or subtract the polynomials.

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

Find the function values for each polynomial function.

For the function $f \left(x\right) = 8 x^{2} - 3 x + 2 ,$ find:
ⓐ $f \left(5\right)$ ⓑ $f \left(−2\right)$ ⓒ $f \left(0\right)$

Find the function values for each polynomial function.

For the function $g \left(x\right) = 4 - 36 x ,$ find:
ⓐ $g \left(3\right)$ ⓑ $g \left(0\right)$ ⓒ $g \left(−1\right)$

Find the height for each polynomial function.

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

Find the height for each polynomial function.

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 function $R \left(p\right) = −4 p^{2} + 420 p .$ Find the revenue received when $p = 60$ dollars.

Find the height for each polynomial function.

The polynomial $C \left(x\right) = 6 x^{2} + 90 x$ gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and height 6 feet. Find the cost of producing a box with $x = 4$ feet.

In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

$f \left(x\right) = 2 x^{2} - 4 x + 1$ and $g \left(x\right) = 5 x^{2} + 8 x + 3$

In each example, find ⓐ (f + g)(x) ⓑ (f + g)(2) ⓒ (f − g)(x) ⓓ (f − g)(−3).

$f \left(x\right) = 3 x^{3} - x^{2} - 2 x + 3$ and $g \left(x\right) = 3 x^{3} - 7 x$

Using your own words, explain the difference between a monomial, a binomial, and a trinomial.

Ariana thinks the sum $6 y^{2} + 5 y^{4}$ is $11 y^{6} .$ What is wrong with her reasoning?