Math  /  Algebra

QuestionFactor the following quadratic expression: 3x2+11x43x^2 + 11x - 4.

Studdy Solution

STEP 1

1. The expression 3x2+11x43x^2 + 11x - 4 is a quadratic polynomial.
2. We are looking for two binomials whose product is the given quadratic expression.
3. The factoring process involves finding two numbers that multiply to the product of the leading coefficient and the constant term, and add to the middle coefficient.

STEP 2

1. Identify the product of the leading coefficient and the constant term.
2. Find two numbers that multiply to this product and add to the middle coefficient.
3. Use these numbers to split the middle term and factor by grouping.
4. Write the expression as a product of two binomials.

STEP 3

Identify the product of the leading coefficient and the constant term. The leading coefficient is 33 and the constant term is 4-4. Calculate the product:
3×(4)=12 3 \times (-4) = -12

STEP 4

Find two numbers that multiply to 12-12 and add to the middle coefficient, which is 1111. These numbers are 1212 and 1-1 because:
12×(1)=12 12 \times (-1) = -12 12+(1)=11 12 + (-1) = 11

STEP 5

Use these numbers to split the middle term 11x11x into two terms: 12x12x and 1x-1x. Rewrite the expression:
3x2+12x1x4 3x^2 + 12x - 1x - 4

STEP 6

Factor by grouping. Group the first two terms and the last two terms:
(3x2+12x)+(1x4) (3x^2 + 12x) + (-1x - 4)
Factor out the greatest common factor from each group:
3x(x+4)1(x+4) 3x(x + 4) - 1(x + 4)

STEP 7

Notice that both groups contain the common factor (x+4)(x + 4). Factor out (x+4)(x + 4):
(3x1)(x+4) (3x - 1)(x + 4)

STEP 8

Write the expression as a product of two binomials. The factored form of the quadratic expression is:
(3x1)(x+4) \boxed{(3x - 1)(x + 4)}
The quadratic expression 3x2+11x43x^2 + 11x - 4 factors to:
(3x1)(x+4) \boxed{(3x - 1)(x + 4)}

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