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 (B)

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

$x^2 + 4x - 21$

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

<p>4</p> Signup and view all the answers

Match the quadratic expression with its factored form:

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

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

<p>8 (C)</p> Signup and view all the answers

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

<p>False (B)</p> Signup and view all the answers

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

<p>$(x+1)(2x-9)$</p> Signup and view all the answers

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

<p>$n^3 - n + 1$</p> Signup and view all the answers

Match the following expressions with their fully factored form:

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

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

<p>$4x^2 + 11x -3$ (A)</p> Signup and view all the answers

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

<p>True (A)</p> Signup and view all the answers

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

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

Flashcards

Factoring

The process of finding two expressions that when multiplied together equal the original expression.

Greatest Common Factor (GCF)

The largest number that divides evenly into two or more numbers. In factoring expressions, it's the greatest common factor of all terms.

Area Model

A visual tool to represent the multiplication of two binomials, showing the individual terms and their products.

Diamond Method

A visual representation of the relationship between the coefficients of a quadratic expression, used to help factor it into two binomials.

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Quadratic Expression

A polynomial expression with an exponent of 2 on the variable.

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Binomial

A polynomial with two terms, often in the form ax + b.

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Trinomial

A polynomial with three terms, often in the form ax^2 + bx + c.

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Standard Form

The form of an expression where terms are arranged from the highest exponent to the lowest.

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Factoring Polynomials with a GCF

To factor an expression completely, first find the greatest common factor (GCF) of all terms. Then, factor out the GCF, leaving the remaining terms in parentheses. Finally, factor the trinomial inside the parentheses if possible.

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Factoring Trinomials

To factor a trinomial, find two numbers that multiply to give the constant term and add to give the coefficient of the middle term.

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Factor x2 + 7x - 30

Finding two numbers that add to 7 and multiply to -30

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Factor x2 - 15x + 56

Finding two numbers that add to -15 and multiply to 56

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Factoring Trinomials with a Leading Coefficient

To factor a trinomial where the leading coefficient is not 1, factor out the GCF first.

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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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