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Factoring Expressions and Quadratics

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Questions and Answers

Which of the following is the correct factorization of $x^2 - 15x + 56$?

$(x - 8)(x - 7)$ (correct)

$(x - 4)(x + 2)$

$(x + 8)(x - 7)$

$(x - 6)(x - 9)$

The expression $2r^2 + 14r + 28$ can be fully factored as $2(r + 2)(r + 7)$?

False

What is the result of expanding $(x - 3)(x + 7)$?

$x^2 + 4x - 21$

The factored form of $x^2 + 6x - 40$ is $(x + 10)(x - ______)$

4 Signup and view all the answers

Match the quadratic expression with its factored form:

$x^2 - 14x + 45$ = $(x - 9)(x - 5)$ $x^2 - 4x - 12$ = $(x - 6)(x + 2)$ $x^2 - 2x - 8$ = $(x - 4)(x + 2)$ $x^2 + 6x - 27$ = $(x + 9)(x - 3)$ Signup and view all the answers

What is the greatest common factor of the expression $16x^3 + 56x + 80$?

8 Signup and view all the answers

The factored form of $x^2 + 13x + 36$ is $(x+6)(x+6)$

False Signup and view all the answers

What is the factored form of $2x^2 -7x - 9$?

$(x+1)(2x-9)$ Signup and view all the answers

The expression $2n^5 - 2n^3 + 2n^2$ factored by the GCF is $2n^2($______$)$.

$n^3 - n + 1$ Signup and view all the answers

Match the following expressions with their fully factored form:

$x^2 - 2x - 24$ = $(x - 6)(x + 4)$ $x^2 - 5x + 6$ = $(x - 3)(x - 2)$ $3x^2 + 20x - 7$ = $(3x - 1)(x + 7)$ $5x^2 - 24x - 5$ = $(x - 5)(5x + 1)$ Signup and view all the answers

Which of the following is the standard form of the area model multiplication $(4x - 1)(x + 3)$?

$4x^2 + 11x -3$ Signup and view all the answers

The factored form of $x^2 + 18x + 77$ is $(x+11)(x+7)$

True Signup and view all the answers

What is the result of the area model $(3x + 2)(2x^2 - 3x + 5)$ in standard form?

$6x^3 - 5x^2 + 9x + 10$ Signup and view all the answers

Study Notes

Factoring Expressions

GCF Factoring: Find the greatest common factor (GCF) of terms in an expression and factor it out. For example, in 30x² - 9x - 6, the GCF is 3, so the factored form is 3(10x² - 3x - 2).

Factoring Quadratics

Area Model: Use an area model to visualize and factor quadratics of the form ax² + bx + c. Break the quadratic into smaller rectangles whose dimensions multiply to the terms. For example, x² + 6x - 16 factors into (x - 2)(x + 8).
Diamond Method: A method for factoring quadratics. This method helps arrange the terms to find factors of the quadratic.
Leading Coefficient: Factoring quadratics with coefficients other than 1, for example, 3x² + 20x – 7, requires careful consideration of the factors that multiply to create the leading coefficient and the constant term.

Factoring Special Cases

Difference of Squares: Factoring expressions like x² - 16. Use the pattern of (a² - b²) = (a-b)(a+b).

Graphing Quadratic Functions

Graphing: Sketching quadratic equations (y = ax² + bx + c). Identify the axis of symmetry, the vertex (maximum or minimum), x-intercepts (zeros), and y-intercept.

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Description

This quiz covers various methods of factoring expressions, including finding the greatest common factor, using area models, and employing the diamond method for quadratics. Additionally, it explores special cases like the difference of squares. Test your understanding of these essential algebraic concepts.