10.1 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.

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

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

1. ⓐ $8 x^{2} - 7 x - 9$
2. ⓑ $−5 a^{4}$
3. ⓒ $x^{4} - 7 x^{3} - 6 x^{2} + 5 x + 2$
4. ⓓ $11 - 4 y^{3}$
5. ⓔ $n$

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

1. ⓐ $z$
2. ⓑ $2 x^{3} - 4 x^{2} - x - 8$
3. ⓒ $6 x^{2} - 4 x + 1$
4. ⓓ $9 - 4 y^{2}$
5. ⓔ $3 x^{7}$

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

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

Find the degree of the following polynomials:

1. ⓐ $4 x$
2. ⓑ $3 x^{3} - 5 x + 7$
3. ⓒ $−11$
4. ⓓ $−6 x^{2} + 9 x - 3$
5. ⓔ $8 x + 2$

Find the degree of the following polynomials:

1. ⓐ $−6 y$
2. ⓑ $4 x - 1$
3. ⓒ $3 x^{4} + 4 x^{2} - 8$
4. ⓓ $2 y^{2} + 3 y + 9$
5. ⓔ $−18$

Find the degree of the following polynomials:

1. ⓐ $47$
2. ⓑ $2 x^{2} - 8 x + 2$
3. ⓒ $x^{4} - 16$
4. ⓓ $y^{5} - 5 y^{3} + y$
5. ⓔ $9 a^{3}$

Add: $17 x^{2} + 6 x^{2} .$

Add: $12 x^{2} + 5 x^{2} .$

Add: $−11 y^{2} + 8 y^{2} .$

Subtract: $11 n - \left(−8 n\right) .$

Subtract: $9 n - \left(−5 n\right) .$

Subtract: $−7 a^{3} - \left(−5 a^{3}\right) .$

Simplify: $a^{2} + 4 b^{2} - 7 a^{2} .$

Add: $3 x^{2} + 3 y^{2} - 5 x^{2} .$

Add: $2 a^{2} + b^{2} - 4 a^{2} .$

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

Find the sum: $\left(7 y^{2} + 4 y - 6\right) + \left(4 y^{2} + 5 y + 1\right) .$

Find the difference: $\left(6 y^{2} + 3 y - 1\right) - \left(3 y^{2} - 4\right) .$

Find the difference: $\left(8 u^{2} - 7 u - 2\right) - \left(5 u^{2} - 6 u - 4\right) .$

Subtract: $\left(4 n^{2} - 7 n - 3\right)$ from $\left(8 n^{2} + 5 n - 3\right) .$

Subtract: $\left(a^{2} - 4 a - 9\right)$ from $\left(6 a^{2} + 4 a - 1\right) .$

Evaluate $3 x^{2} - 9 x + 7$ when

1. ⓐ $x = 3$
2. ⓑ $x = −1$

Evaluate: $2 x^{2} + 4 x - 3$ when

1. ⓐ $x = 2$
2. ⓑ $x = −3$

Evaluate: $7 y^{2} - y - 2$ when

1. ⓐ $y = −4$
2. ⓑ $y = 0$

The polynomial $−8 t^{2} + 24 t + 4$ gives the height, in feet, of a ball $t$ seconds after it is tossed into the air, from an initial height of $4$ feet. Find the height after $t = 3$ seconds.

The polynomial $−8 t^{2} + 24 t + 4$ gives the height, in feet, of a ball $x$ seconds after it is tossed into the air, from an initial height of $4$ feet. Find the height after $t = 2$ seconds.

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

$5 x + 2$

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

$a^{2} + 9 a + 18$

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

$y^{3} - 8 y^{2} + 2 y - 16$

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

$23 y^{2}$

Determine the degree of each polynomial.

$8 a^{5} - 2 a^{3} + 1$

Determine the degree of each polynomial.

$3 x - 12$

Determine the degree of each polynomial.

$−13$

Add or subtract the monomials.

$\text{6x}^{2} + 9 x^{2}$

Add or subtract the monomials.

$−12 u + 4 u$

Add or subtract the monomials.

$5 a + 7 b$

Add or subtract the monomials.

Add: $4 a , - 3 b , - 8 a$

Add or subtract the monomials.

$18 x - 2 x$

Add or subtract the monomials.

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

Add or subtract the polynomials.

$\left(4 y^{2} + 10 y + 3\right) + \left(8 y^{2} - 6 y + 5\right)$

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.

$\left(3 a^{2} + 7\right) + \left(a^{2} - 7 a - 18\right)$

Add or subtract the polynomials.

$\left(6 m^{2} - 9 m - 3\right) - \left(2 m^{2} + m - 5\right)$

Add or subtract the polynomials.

$\left(z^{2} + 8 z + 9\right) - \left(z^{2} - 3 z + 1\right)$

Add or subtract the polynomials.

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

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.

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

Add or subtract the polynomials.

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

evaluate each polynomial for the given value.

$\text{Evaluate} 8 y^{2} - 3 y + 2$

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

evaluate each polynomial for the given value.

$\text{Evaluate} 4 - 36 x \text{when}:$

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

evaluate each polynomial for the given value.

A window washer drops a squeegee from a platform $275$ feet high. The polynomial $−16 t^{2} + 275$ gives the height of the squeegee $t$ seconds after it was dropped. Find the height after $t = 4$ seconds.

Fuel Efficiency The fuel efficiency (in miles per gallon) of a bus going at a speed of $x$ miles per hour is given by the polynomial $- \frac{1}{160} x^{2} + \frac{1}{2} x .$ Find the fuel efficiency when $x = 40 \text{mph}.$

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