Arithmetic with Polynomials and Rational Expressions: Understanding and Operating with Polynomials (CCSS.A-APR.1)
Algebra · Learn by Concept
Help Questions
Algebra › Arithmetic with Polynomials and Rational Expressions: Understanding and Operating with Polynomials (CCSS.A-APR.1)
What is the result when the polynomials are added? $(3x^2 - 2x + 5) + (4x^2 + x - 7)$
$7x^2 + 3x - 2$
$7x^2 - 2$
$7x^2 - x - 2$
$7x^2 - x - 12$
Explanation
Combine like terms: $3x^2 + 4x^2 = 7x^2$, $-2x + x = -x$, and $5 + (-7) = -2$. So the sum is $7x^2 - x - 2$. Choice A makes a sign error on the $x$-term ($-2x + x$ incorrectly treated as $+3x$). Choice B drops the $x$-term. Choice D makes an arithmetic slip with the constants ($5 + (-7)$ incorrectly as $-12$).
What is the product of $(x - 2)(x + 5)$?
$x^2 + 3x - 10$
$x^2 - 3x - 10$
$x^2 - 10$
$(x - 2)(x + 5)$
Explanation
FOIL: $x\cdot x = x^2$, $x\cdot 5 = 5x$, $-2\cdot x = -2x$, $-2\cdot 5 = -10$. Combine: $x^2 + 5x - 2x - 10 = x^2 + 3x - 10$. Choice B flips the sign of the middle term. Choice C drops the outer/inner terms. Choice D leaves the expression factored instead of expanding.
If a rectangle has side lengths $x + 3$ and $x - 4$, what is its area (in square units)?
$x^2 - 7x - 12$
$x^2 - 12$
$x^2 + x - 12$
$x^2 - x - 12$
Explanation
Area is $(x + 3)(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12$. Choice A makes an arithmetic slip combining $-4x$ and $3x$ as $-7x$. Choice B drops the middle term. Choice C has a sign error on the $x$-term.
What is the result when the polynomials are subtracted? $(5x^2 + 2x - 1) - (3x^2 - 4x + 6)$
$2x^2 - 2x + 5$
$2x^2 + 6x - 7$
$2x^2 + 8x - 7$
$2x^2 - 7$
Explanation
Distribute the negative: $(5x^2 + 2x - 1) + (-3x^2 + 4x - 6)$. Combine: $(5x^2 - 3x^2) = 2x^2$, $(2x + 4x) = 6x$, $(-1 - 6) = -7$. Choice A forgets to distribute the negative to both terms in the second polynomial. Choice C makes an arithmetic slip on the $x$-terms. Choice D drops the $x$-term entirely.
Compute the product: $(2x + 3)(x^2 - x + 2)$
$2x^3 + x^2 + x + 6$
$2x^3 + 5x^2 + x + 6$
$2x^3 + x^2 + 7x + 6$
$2x^3 + x^2 + 4x + 6$
Explanation
Distribute: $2x(x^2 - x + 2) = 2x^3 - 2x^2 + 4x$ and $3(x^2 - x + 2) = 3x^2 - 3x + 6$. Combine like terms: $2x^3 + (-2x^2 + 3x^2) + (4x - 3x) + 6 = 2x^3 + x^2 + x + 6$. Choice B makes a sign error on $-2x^2$ (treated as $+2x^2$). Choice C makes an arithmetic slip combining $4x$ and $-3x$ as $7x$. Choice D drops the $-3x$ term.
What is the result when the polynomials are added? $(3x^2 + 2x - 5) + (x^2 - 4x + 7)$
$4x^2 - 2x + 2$
$4x^2 + 2x + 2$
$4x^2 - 2x + 12$
$4x^2 - 2x$
Explanation
Combine like terms: $(3x^2 + x^2) = 4x^2$, $(2x - 4x) = -2x$, and $(-5 + 7) = 2$, so the sum is $4x^2 - 2x + 2$. Choice B has a sign error on the $x$ term; Choice C mis-adds the constants; Choice D drops the constant term.
If a rectangle has sides $(x + 3)$ and $(x - 4)$, what is its area?
$x^2 - 12$
$x^2 - x - 12$
$x^2 - 7x + 12$
$(x + 3)(x - 4)$
Explanation
Multiply the binomials: $x(x - 4) + 3(x - 4) = x^2 - 4x + 3x - 12 = x^2 - x - 12$. Choice A drops the middle term; Choice C has a sign error on the constant and an incorrect middle coefficient; Choice D is left unexpanded.
A business has revenue $R(x) = 6x + 40$ and cost $C(x) = 4x + 25$. What is the profit $P(x) = R(x) - C(x)$?
$10x + 65$
$2x + 65$
$10x + 15$
$2x + 15$
Explanation
Subtract: $(6x - 4x) + (40 - 25) = 2x + 15$. Choice A adds instead of subtracting; Choice B miscomputes $40 - 25$; Choice C adds the $x$-coefficients as if adding functions.
What is the product of $(x + 3)(x^2 - x + 2)$?
$x^3 + 2x^2 - x + 6$
$x^3 + 2x^2 + x + 6$
$x^3 + x^2 - x + 6$
$(x + 3)(x^2 - x + 2)$
Explanation
Distribute: $x(x^2 - x + 2) = x^3 - x^2 + 2x$ and $3(x^2 - x + 2) = 3x^2 - 3x + 6$. Combine: $x^3 + 2x^2 - x + 6$. Choice B flips the sign of the linear term; Choice C misadds the $x^2$ terms; Choice D is not expanded.