Algebra Class: Simplifying and Factoring Polynomials

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Questions and Answers

What is the simplified form of the expression (3b + 5 − 4b^2) − (8b^2 + 6b − 3)?

−5b^2 + 3b + 8
−5b^2 + 9
−12b^2 + 3b + 8
−12b^2 − 3b + 8 (correct)

What is the result of multiplying (4x − 4)(4x + 4)?

16x^2 + 16
16x^2 + 8
16x^2 − 16 (correct)
16x^2 − 8

Which polynomial term describes 4v^3 + 6?

Monomial
Cubic monomial
Cubic binomial (correct)
Quadratic binomial

What is the expression resulting from (6 + 8x^2)(6 − 8x^2)?

36 − 64x^4 (B) Signup and view all the answers

What is the result of combining (6r^3 − r + 8r^4) + (4r^4 + 8r^3 + 2r)?

12r^4 + 14r^3 + r (B) Signup and view all the answers

What is the product of (6a - 3)^2?

36a^2 − 36a + 9 (B) Signup and view all the answers

How would you classify the term −8r^3?

Cubic monomial (A) Signup and view all the answers

What is the value of the simplified expression (8k − 3)(8k + 3)?

64k^2 − 9 (C) Signup and view all the answers

What is the factored form of the expression $m^2 + 6m + 9$?

$(m + 3)(m + 3)$ (B) Signup and view all the answers

Which expression corresponds to the difference of squares?

$100x^2 - 36$ (D), $4 - 324m^2$ (A), $25n^2 - 16$ (B) Signup and view all the answers

Identify the correct factored form of $x^2 - 4x - 5$.

$(x - 5)(x + 1)$ (D) Signup and view all the answers

What is the value of the coefficient of $n$ in the expression $20n^2 - 100n + 125$?

-100 (B) Signup and view all the answers

What is the correct factorization of the expression $p^2 - 3p + 2$?

$(p - 1)(p - 2)$ (A) Signup and view all the answers

Which expression has a double root?

$n^2 + 10n + 25$ (A), $m^2 - 8m + 16$ (B) Signup and view all the answers

What is the result of factoring the expression $4p^2 - 20p + 25$?

$(2p - 5)(2p - 5)$ (A) Signup and view all the answers

What is the correct factorization of $25x^2 - 1$?

$(5x - 1)(5x + 1)$ (B) Signup and view all the answers

What is the simplified form of $(3x + 3x^3 - 5x^4) + (4x^3 + 3x - 5x^4)$?

$7x^3 - 10x^4$ (B) Signup and view all the answers

What is the product of $(6 + 8x^2)(6 - 8x^2)$?

$36 - 64x^4$ (B) Signup and view all the answers

What describes the polynomial $4v^3 + 6$ in terms of degree and number of terms?

Degree 3, 2 terms (A) Signup and view all the answers

What is the factored form of $36x^2y^5 + 54y^2 - 12$?

$6(6xy^5 + 9y - 2)$ (D) Signup and view all the answers

What is the simplified result of $(3b + 5 - 4b^2) - (8b^2 + 6b - 3)$?

$-12b^2 - 3b + 8$ (D) Signup and view all the answers

What is the completely factored form of $-3m - 3mn^3 + 3m^4n^2$?

$-3m(1 + mn^3 - m^3n^2)$ (A) Signup and view all the answers

What is the product of $(8k - 3)(8k + 3)$?

$64k^2 - 9$ (C) Signup and view all the answers

How many terms does the polynomial $-8r^3$ have?

1 term (A) Signup and view all the answers

What is the completely factored form of the expression $36x^2y^5 + 54y^2 - 12$?

$6(6x^2y^5 + 9y^2 - 2)$ (D) Signup and view all the answers

Which expression is the completely factored form of $-3m - 3mn^3 + 3m^4n^2$?

$3m(-1 - n^3 + m^3n^2)$ (D) Signup and view all the answers

What is the factored form of $42xy^4 + 7x^3y + 70xy$?

$7xy(6y^3 + x^2 + 10)$ (A) Signup and view all the answers

What is the completely factored form of the quadratic expression $m^2 + 6m + 9$?

$(m + 3)^2$ (D) Signup and view all the answers

What is the factored form of $p^2 - 8p + 16$?

$(p - 4)^2$ (B) Signup and view all the answers

What is the factored form of the expression $4 - 324m^2$?

$4(1 + 9m)(1 - 9m)$ (A) Signup and view all the answers

Which expression represents the completely factored form of $25n^2 - 16$?

$(5n + 4)(5n - 4)$ (C) Signup and view all the answers

What is the completely factored form of $20x^2 + 100x + 125$?

$5(2x + 5)^2$ (D) Signup and view all the answers

What is the factored form of the polynomial $x^2 - 4x - 5$?

$(x + 1)(x - 5)$ (D) Signup and view all the answers

What is the factored form of $m^2 - 3m - 10$?

$(m - 5)(m + 2)$ (D) Signup and view all the answers

What is the completely factored form of $n^2 + 10n + 25$?

$(n + 5)^2$ (B) Signup and view all the answers

Which is the factored form of $4p^2 - 20p + 25$?

$(2p - 5)^2$ (A) Signup and view all the answers

What is the factored form of $20n^2 - 100n + 125$?

$5(2n - 5)$ (B) Signup and view all the answers

What is the completely factored form of $k^2 - 4$?

$(k + 2)(k - 2)$ (B) Signup and view all the answers

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Study Notes

Simplifying Expressions

The sum of (3x + 3x³ - 5x⁴) and (4x³ + 3x - 5x⁴) is -10x⁴ + 7x³ + 6x
The sum of (6r³ - r + 8r⁴) and (4r⁴ + 8r³ + 2r) is 12r⁴ + 14r³ + r
The difference between (3b + 5 - 4b²) and (8b² + 6b - 3) is -12b² - 3b + 8
The sum of (8v + 1 + 8v³) and (5v⁴ - 3v³ - 6) is 5v⁴ + 5v³ + 8v - 5

Finding Products

The product of (4x - 4) and (4x + 4) is 16x² - 16
The product of (6 + 8x²) and (6 - 8x²) is 36 - 64x⁴
The square of (6a - 3) is 36a² - 36a + 9
The product of (8k - 3) and (8k + 3) is 64k² - 9

Polynomial Classification

-1 is a constant monomial.
-8r³ is a cubic monomial
-8 is a constant monomial
4v³ + 6 is a cubic binomial

Factoring Common Factors

The common factor of 36x²y⁵ + 54y² - 12 is 6, which results in 6( 6x²y⁵ + 9y² - 2)
The common factor of -3m - 3mn³ + 3m⁴n² is 3m, which results in 3m(-1 - n³ + m³n²)
The common factor of 42xy⁴ + 7x³y + 70xy is 7xy, which results in 7xy(6y³ + x² + 10)
The common factor of -36x³y² + 4x³y - 16x²y is 4x²y, which results in 4x²y(-9xy + x - 4)

Factoring Completely

The factored form of m² + 6m + 9 is (m + 3)²
The factored form of 5n² - 40n + 80 is 5(n - 4)²
The factored form of p² - 8p + 16 is (p - 4)²
The factored form of 4 - 324m² is 4(1 + 9m)(1 - 9m)
The factored form of m² + 6m - 27 is (m + 9)(m - 3)
The factored form of m² - 3m - 10 is (m - 5)(m + 2)
The factored form of p² - 3p + 2 is (p - 2)(p - 1)
The factored form of r² - 15r + 56 is (r - 8)(r - 7)
The factored form of x² - 4x - 5 is (x + 1)(x - 5)
The factored form of x² + 8x is x(x + 8)
The factored form of n² + 10n + 25 is (n + 5)²
The factored form of n² - 13n + 30 is (n - 10)(n - 3)
The factored form of 20x² + 100x + 125 is 5(2x + 5)²
The factored form of k² - 4 is (k + 2)(k - 2)
The factored form of 25n² - 16 is (5n + 4)(5n - 4)
The factored form of 100x² - 36 is 4(5x + 3)(5x - 3)
The factored form of 25x² - 1 is (5x+ 1)(5x - 1)
The factored form of m² - 8m + 16 is (m - 4)²
The factored form of 4p² - 20p + 25 is (2p - 5)²
The factored form of 20n² - 100n + 125 is 5(2n - 5)²

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