Math Problem Statement

answer

Factor the following polynomials: a. 3x^2 - 6x + 3, b. -4x^3 + 2x^2 + 6x, c. 15x^2 + 7x - 2, d. x^3 - x + 2x^2 - 2

Solution

Let's solve the given problems by factoring the provided polynomials.

a. 3x26x+33x^2 - 6x + 3

We can factor out the greatest common factor (GCF) of all terms: 3x26x+3=3(x22x+1)3x^2 - 6x + 3 = 3(x^2 - 2x + 1) Now, factor the quadratic inside the parentheses: x22x+1=(x1)(x1)=(x1)2x^2 - 2x + 1 = (x - 1)(x - 1) = (x - 1)^2 Thus, the factored form is: 3(x1)23(x - 1)^2

b. 4x3+2x2+6x-4x^3 + 2x^2 + 6x

First, factor out the greatest common factor (GCF): 4x3+2x2+6x=2x(2x2x3)-4x^3 + 2x^2 + 6x = -2x(2x^2 - x - 3) Now, we factor the quadratic 2x2x32x^2 - x - 3. Look for two numbers that multiply to 2(3)=62(-3) = -6 and add to 1-1. These numbers are 3-3 and 22. Rewrite the middle term: 2x23x+2x3=(2x23x)+(2x3)2x^2 - 3x + 2x - 3 = (2x^2 - 3x) + (2x - 3) Factor by grouping: =x(2x3)+1(2x3)=(2x3)(x+1)= x(2x - 3) + 1(2x - 3) = (2x - 3)(x + 1) Thus, the factored form is: 2x(2x3)(x+1)-2x(2x - 3)(x + 1)

c. 15x2+7x215x^2 + 7x - 2

We need to factor this quadratic. Find two numbers that multiply to 15(2)=3015(-2) = -30 and add to 77. These numbers are 1010 and 3-3. Rewrite the middle term: 15x2+10x3x2=(15x2+10x)(3x+2)15x^2 + 10x - 3x - 2 = (15x^2 + 10x) - (3x + 2) Factor by grouping: =5x(3x+2)1(3x+2)=(3x+2)(5x1)= 5x(3x + 2) - 1(3x + 2) = (3x + 2)(5x - 1) Thus, the factored form is: (3x+2)(5x1)(3x + 2)(5x - 1)

d. x3x+2x22x^3 - x + 2x^2 - 2

First, rearrange the terms in standard form: x3+2x2x2x^3 + 2x^2 - x - 2 Now, factor by grouping: (x3+2x2)(x+2)=x2(x+2)1(x+2)(x^3 + 2x^2) - (x + 2) = x^2(x + 2) - 1(x + 2) Factor out the common binomial factor: =(x21)(x+2)= (x^2 - 1)(x + 2) Notice that x21x^2 - 1 is a difference of squares: x21=(x1)(x+1)x^2 - 1 = (x - 1)(x + 1) Thus, the factored form is: (x1)(x+1)(x+2)(x - 1)(x + 1)(x + 2)

Would you like further details on any of these factorizations or have any questions?

Here are five related questions:

  1. How do you factor a quadratic trinomial?
  2. What are the steps for factoring by grouping?
  3. How do you recognize a difference of squares?
  4. Can you always factor a cubic polynomial?
  5. What is the significance of finding the greatest common factor (GCF) first?

Tip: Always check for the GCF before trying more advanced factoring techniques. It simplifies the polynomial and makes the process easier.

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Math Problem Analysis

Mathematical Concepts

Factoring Polynomials
Greatest Common Factor (GCF)
Factoring by Grouping
Difference of Squares

Formulas

ax^2 + bx + c = 0 (Quadratic Formula)
Factoring by Grouping: ax^2 + bx + c → (ax + p)(bx + q)
Difference of Squares: a^2 - b^2 = (a - b)(a + b)

Theorems

Quadratic Theorem
Difference of Squares Theorem
Greatest Common Factor (GCF) Method

Suitable Grade Level

Grades 9-12