This algebra 1 polynomials worksheet answer key solves all 23 problems of the adding, subtracting and multiplying polynomials worksheet step by step.
Every solution below shows the steps: the combined like terms, the four FOIL products and the checks. Compare each step with your own work, not only the final answer in bold.
Algebra 1 Polynomials Worksheet Answer Key: problems 1–5
Problem 1: Degree and terms
The largest exponent is 3, so the degree is 3. The term \(4x^{3}\) has the largest exponent, so the leading coefficient is 4. There are 3 terms, so it is a trinomial.
Problem 2: Standard form
Order the terms from the highest power to the lowest, keeping each sign with its term: \(x^{3}\), \(-2x^{2}\), \(6x\), \(5\).
\(x^{3} – 2 x^{2} + 6 x + 5\)
Problem 3: Classify polynomials
- One term, degree 2: a monomial of degree 2.
- Two terms, degree 2: a binomial of degree 2.
- Three terms, degree 3: a trinomial of degree 3.
Problem 4: Add polynomials
Combine like terms column by column: \(3x^{2}+2x^{2}=5x^{2}\), \(5x-8x=-3x\), \(-4+9=5\).
\(5 x^{2} – 3 x + 5\)
Problem 5: Subtract polynomials
Distribute the negative sign: \(7x^{2}-3x+1-4x^{2}-2x+6\).
Combine like terms: \((7-4)x^{2}+(-3-2)x+(1+6)\).
\(3 x^{2} – 5 x + 7\)
Algebra 1 Polynomials Worksheet Answer Key: problems 6–10
Problem 6: Two variables
Match like terms: \(x^{2}+5x^{2}=6x^{2}\), \(4xy-2xy=2xy\), \(-3y^{2}+y^{2}=-2y^{2}\).
\(6 x^{2} + 2 x y – 2 y^{2}\)
Problem 7: Subtract with a missing term
Distribute the negative: \(6a^{3}-2a+5-a^{3}-4a^{2}+2a+3\).
The \(a\) terms cancel: \(-2a+2a=0\). The constants give \(5+3=8\).
\(5 a^{3} – 4 a^{2} + 8\)
Problem 8: Multiply monomials
Multiply the coefficients: \(3\cdot(-4)=-12\). Add the exponents: \(x^{2}\cdot x^{5}=x^{7}\).
\(- 12 x^{7}\)
Problem 9: Distribute a monomial
\(5x\cdot 2x^{2}=10x^{3}\), \(5x\cdot(-3x)=-15x^{2}\), \(5x\cdot 4=20x\).
\(10 x^{3} – 15 x^{2} + 20 x\)
Problem 10: Box method
The cells are \(x\cdot x=x^{2}\), \(4x\), \(3x\) and \(12\). Combine \(4x+3x=7x\).
\(x^{2} + 7 x + 12\)
Algebra 1 Polynomials Worksheet Answer Key: problems 11–15
Problem 11: FOIL with a negative
First: \(2 x^{2}\). Outer: \(12 x\). Inner: \(- 5 x\). Last: \(-30\).
Add: \(2 x^{2} + 12 x – 5 x – 30\), so \(12x-5x=7x\).
\(2 x^{2} + 7 x – 30\)
Problem 12: FOIL, both negative
First: \(12 x^{2}\). Outer: \(- 21 x\). Inner: \(- 8 x\). Last: \(14\).
Combine \(-21x-8x=-29x\).
\(12 x^{2} – 29 x + 14\)
Problem 13: Square of a binomial
Use \((a+b)^{2}=a^{2}+2ab+b^{2}\) with \(a=x\) and \(b=7\): \(x^{2}+2(x)(7)+7^{2}\).
\(x^{2} + 14 x + 49\)
Problem 14: Square of a difference
Use \((a-b)^{2}=a^{2}-2ab+b^{2}\) with \(a=3x\) and \(b=5\): \((3x)^{2}-2(3x)(5)+5^{2}\).
\(9 x^{2} – 30 x + 25\)
Problem 15: Difference of squares
The middle terms cancel, so \((a+b)(a-b)=a^{2}-b^{2}\) with \(a=2x\) and \(b=9\).
\(4 x^{2} – 81\)
Algebra 1 Polynomials Worksheet Answer Key: problems 16–20
Problem 16: Binomial times trinomial
Distribute \(x\): \(x^{3}+2x^{2}-5x\). Distribute \(-3\): \(-3x^{2}-6x+15\).
Combine like terms: \(2x^{2}-3x^{2}=-x^{2}\) and \(-5x-6x=-11x\).
\(x^{3} – x^{2} – 11 x + 15\)
Problem 17: Closure
The \(x^{2}\) terms cancel: \(4x^{2}-4x^{2}=0\). The sum is \(3x+5\), with degree 1.
\(3 x + 5\) is still a polynomial, which agrees with closure under addition. Note that the sum can have a lower degree than either addend.
Problem 18: Garden area
Area = length times width: \((x+8)(x+5)=x^{2}+5x+8x+40\), so the area is \(x^{2} + 13 x + 40\) square feet.
For \(x=4\): \(16+52+40=108\). Check with the dimensions: \(12\times 9=108\).
108 square feet
Problem 19: Profit polynomial
\(P=(-x^{2}+50x)-(10x+120)=-x^{2}+50x-10x-120\), so \(- x^{2} + 40 x – 120\).
For \(x=10\): \(-100+400-120=180\). Check: revenue \(\$400\) minus cost \(\$220\) is \(\$180\).
\$180
Problem 20: Picture frame
Border = outer square minus photo: \((x+8)^{2}-(x+2)^{2}\).
\((x^{2}+16x+64)-(x^{2}+4x+4)=12x+60\).
For \(x=4\): \(12(4)+60=108\). Check: \(12^{2}-6^{2}=144-36=108\).
\(12 x + 60\); 108 square inches
Algebra 1 Polynomials Worksheet Answer Key: problems 21–23
Problem 21: Mental math
Write \(52=50+2\) and \(48=50-2\). Then \((50+2)(50-2)=50^{2}-2^{2}\) by the difference of squares pattern.
\(2500-4=2496\).
2,496
Problem 22: Metal sheet with a cutout
Remaining area = sheet minus square: \((3x+4)(x+2)-x^{2}\).
\((3x^{2}+6x+4x+8)-x^{2}=2x^{2}+10x+8\).
For \(x=3\): \(18+30+8=56\). Check: \(13\times 5-3^{2}=65-9=56\).
\(2 x^{2} + 10 x + 8\); 56 square inches
Problem 23: Find the error
At \(x=1\): the left side is \((1+5)^{2}=36\) but Noah’s right side is \(1+25=26\). They differ, so he is wrong.
He squared each term separately. Squaring means multiplying the binomial by itself, so the middle term \(2\cdot x\cdot 5=10x\) is missing.
\(x^{2} + 10 x + 25\)
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Read this algebra 1 polynomials worksheet answer key after trying each problem yourself, and redo any problem you missed.
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Algebra 1 Adding, Subtracting and Multiplying Polynomials worksheet answer key
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