These 23 problems practice naming and writing polynomials, adding and subtracting them, and multiplying with the distributive property, FOIL, the box method and the special products (Common Core A-APR.A.1 and A-SSE.A.2).
The problems start with degree and standard form and get harder. The last six are word problems with areas, profit and mental math.
Problem 1: Degree and terms
For \(4 x^{3} – 7 x + 2\), state the degree, the leading coefficient and the number of terms. Then classify it as a monomial, binomial or trinomial.
Problem 2: Standard form
Write \(5-2x^{2}+6x+x^{3}\) in standard form.
Problem 3: Classify polynomials
Name each polynomial by its number of terms and its degree.
- \(7x^{2}\)
- \(x^{2}-9\)
- \(2x^{3}+x-1\)
Problem 4: Add polynomials
Add \((3x^{2}+5x-4)+(2x^{2}-8x+9)\).
Problem 5: Subtract polynomials
Subtract \((7x^{2}-3x+1)-(4x^{2}+2x-6)\).
Problem 6: Two variables
Add \((x^{2}+4xy-3y^{2})+(5x^{2}-2xy+y^{2})\).
Problem 7: Subtract with a missing term
Subtract \((6a^{3}-2a+5)-(a^{3}+4a^{2}-2a-3)\) and write the answer in standard form.
Problem 8: Multiply monomials
Multiply \((3x^{2})(-4x^{5})\).
Problem 9: Distribute a monomial
Multiply \(5x(2x^{2}-3x+4)\).
Problem 10: Box method
Use the area model to multiply \((x+4)(x+3)\).
Problem 11: FOIL with a negative
Multiply \((2x-5)(x+6)\).
Problem 12: FOIL, both negative
Multiply \((3x-2)(4x-7)\).
Problem 13: Square of a binomial
Expand \((x+7)^{2}\). The figure shows a square with side \(x+7\).
Problem 14: Square of a difference
Expand \((3x-5)^{2}\).
Problem 15: Difference of squares
Multiply \((2x+9)(2x-9)\).
Problem 16: Binomial times trinomial
Multiply \((x-3)(x^{2}+2x-5)\).
Problem 17: Closure
Add \((4x^{2}+3x)+(-4x^{2}+5)\). State the degree of the sum and explain whether the result is still a polynomial.
Problem 18: Garden area
Maya plans a rectangular garden that is \(x+8\) feet long and \(x+5\) feet wide. Write a polynomial for its area, then find the area when \(x=4\).
Problem 19: Profit polynomial
The school store sells \(x\) dozen cookies. Revenue is \(R=-x^{2}+50x\) dollars and cost is \(C=10x+120\) dollars. Write a polynomial for the profit \(P=R-C\), then find the profit for 10 dozen cookies.
Problem 20: Picture frame
Jamal puts a square photo with side \(x+2\) inches into a square frame with outer side \(x+8\) inches. Write a simplified polynomial for the area of the frame itself (the border), then find it when \(x=4\).
Problem 21: Mental math
Priya calculates \(52\times 48\) in her head by writing it as a product of a sum and a difference. Show her steps and find the product.
Problem 22: Metal sheet with a cutout
A metal sheet is \(3x+4\) inches long and \(x+2\) inches wide. A square with side \(x\) inches is cut from one corner. Write a simplified polynomial for the area that remains, then evaluate it for \(x=3\).
Problem 23: Find the error
Noah writes \((x+5)^{2}=x^{2}+25\). Test his answer with \(x=1\), explain his mistake and write the correct expansion.
See every chapter on the Algebra 1 math page.