Adding and Subtracting Polynomials

Show your work for each problem.

1.
Which terms are like terms?
1. a) $3x^2$ and $5x^2$
2. b) $2x$ and $2y$
3. c) $x^3$ and $x^2$
4. d) $3x^2$ and $3x$
Answer: $3x^2$ and $5x^2$
$3x^2$ and $5x^2$ are like terms because both have the variable $x$ raised to the power 2. The coefficients (3 and 5) can be different.
2.
Combine the like terms: $4x^2 + 7x^2 = \square\, x^2$ . What number goes in the box?
Answer: 11
$4x^2 + 7x^2 = (4 + 7)x^2 = 11x^2$ . Add the coefficients, keep the same variable and exponent. The box is $11$ .
3.
Combine the like terms: $9x - 4x = \square\, x$ . What number goes in the box?
Answer: 5
$9x - 4x = (9 - 4)x = 5x$ . Subtract the coefficients when the operation is subtraction. The box is $5$ .
4.
What is the result of $(2x + 5) + (3x + 4)$ ?
1. a) $6x + 9$
2. b) $5x^2 + 9$
3. c) $5x + 1$
4. d) $5x + 9$
Answer: $5x + 9$
$(2x + 5) + (3x + 4) = (2x + 3x) + (5 + 4) = 5x + 9$
5.
Add: $(3x^2 + 2x) + (x^2 + 5x)$
Answer: 4x^2 + 7x
1. Add the $x^2$ terms: $3x^2 + x^2$ 4x^2
2. Add the $x$ terms: $2x + 5x$ 7x
3. Write the final answer 4x^2 + 7x
6.
What is $(5x^2 - 3x + 2) - (2x^2 + x - 4)$ ?
1. a) $3x^2 - 4x - 2$
2. b) $7x^2 - 2x - 2$
3. c) $3x^2 - 2x - 2$
4. d) $3x^2 - 4x + 6$
Answer: $3x^2 - 4x + 6$
$(5x^2 - 3x + 2) - (2x^2 + x - 4) = 5x^2 - 3x + 2 - 2x^2 - x + 4 = 3x^2 - 4x + 6$
7.
Add: $(x^2 + 4x - 3) + (2x^2 - x + 7)$ . What is the constant term of the result?
Answer: 4
$(x^2 + 4x - 3) + (2x^2 - x + 7) = 3x^2 + 3x + 4$ . We get: $1+2=3$ for $x^2$ , $4-1=3$ for $x$ , and $-3+7=4$ for constants. The constant term is $4$ .
8.
Subtract: $(4x^2 + 5x - 1) - (x^2 - 3x + 2)$
Answer: 3x^2 + 8x - 3
1. Distribute the negative: $-(x^2 - 3x + 2) = ?$ -x^2 + 3x - 2
2. Combine $x^2$ terms: $4x^2 + (-x^2)$ 3x^2
3. Combine $x$ terms: $5x + 3x$ 8x
4. Combine constants: $-1 + (-2)$ -3
5. Write the final polynomial 3x^2 + 8x - 3
9.
Add: $(x^3 + 2x) + (4x^2 - x + 5)$ . What is the coefficient of $x$ in the result?
Answer: 1
$(x^3 + 2x) + (4x^2 - x + 5) = x^3 + 4x^2 + (2x - x) + 5 = x^3 + 4x^2 + x + 5$ . Write in standard form (descending powers). The $x$ term is $x = 1x$ , so the coefficient is $1$ .
10.
Simplify: $(6x^2 - 4x + 3) - (2x^2 - 4x - 5)$
Answer: 4x^2 + 8
1. Distribute the negative: $-(2x^2 - 4x - 5)$ -2x^2 + 4x + 5
2. Combine $x^2$ terms: $6x^2 - 2x^2$ 4x^2
3. Combine $x$ terms: $-4x + 4x$ 0
4. Combine constants: $3 + 5$ 8
5. Write the final answer 4x^2 + 8
11.
A rectangle has length $(3x + 5)$ and width $(x + 2)$ . If the length is increased by $(x - 1)$ , what is the new length?
1. a) $3x + 4$
2. b) $4x + 4$
3. c) $4x + 6$
4. d) $2x + 6$
Answer: $4x + 4$
New length $= (3x + 5) + (x - 1) = 3x + x + 5 - 1 = 4x + 4$
12.
Subtract: $(3x^2 - 2x + 1) - (-x^2 + 4x - 3)$ . What is the coefficient of $x$ in the result?
Answer: -6
$(3x^2 - 2x + 1) - (-x^2 + 4x - 3) = 3x^2 - 2x + 1 + x^2 - 4x + 3 = 4x^2 - 6x + 4$ . The coefficient of $x$ is $-2 - 4 = -6$ .
13.
Simplify: $(2x^3 - x^2 + 3x) + (x^3 + 4x^2 - x + 2) - (x^3 - 2x + 1)$
Answer: 2x^3 + 3x^2 + 4x + 1
1. Distribute the negative to the last polynomial: $-(x^3 - 2x + 1)$ -x^3 + 2x - 1
2. Combine all $x^3$ terms: $2x^3 + x^3 - x^3$ 2x^3
3. Combine all $x^2$ terms: $-x^2 + 4x^2$ 3x^2
4. Combine all $x$ terms: $3x - x + 2x$ 4x
5. Combine constants: $2 - 1$ 1
6. Write the final answer in standard form 2x^3 + 3x^2 + 4x + 1
14.
If $P(x) = 5x^2 - 3x + 7$ and $Q(x) = 2x^2 + x - 4$ , what is $P(x) - 2Q(x)$ ?
1. a) $x^2 - x + 15$
2. b) $x^2 - 5x - 1$
3. c) $x^2 - 5x + 15$
4. d) $9x^2 - 5x - 1$
Answer: $x^2 - 5x + 15$
$P(x) - 2Q(x) = (5x^2 - 3x + 7) - 2(2x^2 + x - 4) = 5x^2 - 3x + 7 - 4x^2 - 2x + 8 = x^2 - 5x + 15$
15.
Find the polynomial that must be added to $(2x^2 - 5x + 3)$ to get $(5x^2 + x - 4)$ .
Answer: 3x^2 + 6x - 7
1. Set up the equation: $(2x^2 - 5x + 3) + ? = (5x^2 + x - 4)$ . What operation do we need? subtraction
2. Rewrite as subtraction: $(5x^2 + x - 4) - (2x^2 - 5x + 3)$ . Distribute the negative. -2x^2 + 5x - 3
3. Combine $x^2$ terms: $5x^2 - 2x^2$ 3x^2
4. Combine $x$ terms: $x + 5x$ 6x
5. Combine constants: $-4 - 3$ -7
6. Write the missing polynomial 3x^2 + 6x - 7
16.
Work out: $-(3x^2 - 2x + 4) + (x^2 + 5x - 1) - (2x^2 - x)$ . What is the coefficient of $x^2$ in the result?
Answer: -4
$-(3x^2 - 2x + 4) + (x^2 + 5x - 1) - (2x^2 - x) = -3x^2 + 2x - 4 + x^2 + 5x - 1 - 2x^2 + x = -4x^2 + 8x - 5$ . The coefficient of $x^2$ is $-3 + 1 - 2 = -4$ .