Question

factor. check by using foil.
4
$6x^2 - 19x - 7$
5
$6x^2 - x - 7$
6
$6x^2 + 41x - 7$

Explanation:

Problem 4: Factor \(6x^2 - 19x - 7\) and check by FOIL

Step 1: Find \(a \times c\)

For \(ax^2 + bx + c\), here \(a = 6\), \(b = -19\), \(c = -7\). So \(a \times c = 6 \times (-7) = -42\). We need two numbers that multiply to \(-42\) and add to \(b = -19\). The numbers are \(-21\) and \(2\) (since \(-21 \times 2 = -42\) and \(-21 + 2 = -19\)).

Step 2: Rewrite the middle term

Split \(-19x\) using \(-21x\) and \(2x\):
\(6x^2 - 21x + 2x - 7\)

Step 3: Group and factor

Group the first two and last two terms:
\((6x^2 - 21x) + (2x - 7)\)
Factor out the GCF from each group:
\(3x(2x - 7) + 1(2x - 7)\)

Step 4: Factor out the common binomial

Factor out \((2x - 7)\):
\((3x + 1)(2x - 7)\)

Step 5: Check by FOIL

Multiply \((3x + 1)(2x - 7)\):
First: \(3x \times 2x = 6x^2\)
Outer: \(3x \times (-7) = -21x\)
Inner: \(1 \times 2x = 2x\)
Last: \(1 \times (-7) = -7\)
Combine like terms: \(6x^2 - 21x + 2x - 7 = 6x^2 - 19x - 7\), which matches the original.

Step 1: Find \(a \times c\)

\(a = 6\), \(b = -1\), \(c = -7\). So \(a \times c = 6 \times (-7) = -42\). We need two numbers that multiply to \(-42\) and add to \(-1\). The numbers are \(-7\) and \(6\) (since \(-7 \times 6 = -42\) and \(-7 + 6 = -1\)).

Step 2: Rewrite the middle term

Split \(-x\) using \(-7x\) and \(6x\):
\(6x^2 - 7x + 6x - 7\)

Step 3: Group and factor

Group the first two and last two terms:
\((6x^2 - 7x) + (6x - 7)\)
Factor out the GCF from each group:
\(x(6x - 7) + 1(6x - 7)\)

Step 4: Factor out the common binomial

Factor out \((6x - 7)\):
\((x + 1)(6x - 7)\)

Step 5: Check by FOIL

Multiply \((x + 1)(6x - 7)\):
First: \(x \times 6x = 6x^2\)
Outer: \(x \times (-7) = -7x\)
Inner: \(1 \times 6x = 6x\)
Last: \(1 \times (-7) = -7\)
Combine like terms: \(6x^2 - 7x + 6x - 7 = 6x^2 - x - 7\), which matches the original.

Step 1: Find \(a \times c\)

\(a = 6\), \(b = 41\), \(c = -7\). So \(a \times c = 6 \times (-7) = -42\). We need two numbers that multiply to \(-42\) and add to \(41\). The numbers are \(42\) and \(-1\) (since \(42 \times (-1) = -42\) and \(42 + (-1) = 41\)).

Step 2: Rewrite the middle term

Split \(41x\) using \(42x\) and \(-x\):
\(6x^2 + 42x - x - 7\)

Step 3: Group and factor

Group the first two and last two terms:
\((6x^2 + 42x) + (-x - 7)\)
Factor out the GCF from each group:
\(6x(x + 7) - 1(x + 7)\)

Step 4: Factor out the common binomial

Factor out \((x + 7)\):
\((6x - 1)(x + 7)\)

Step 5: Check by FOIL

Multiply \((6x - 1)(x + 7)\):
First: \(6x \times x = 6x^2\)
Outer: \(6x \times 7 = 42x\)
Inner: \(-1 \times x = -x\)
Last: \(-1 \times 7 = -7\)
Combine like terms: \(6x^2 + 42x - x - 7 = 6x^2 + 41x - 7\), which matches the original.

Answer:

\((3x + 1)(2x - 7)\)

Problem 5: Factor \(6x^2 - x - 7\) and check by FOIL