Question

factoring trinomials (ax² + bx + c)
how do you factor a trinomial in the form ax² + bx + c?
list all the ways you can
to the product of “a” and “c”.
determine which of those combinations also to “b”.
(note: sometimes one or both factors are negative.
test the negative signs once determining the factors)
guided example: factor 2x² + 15x + 18.
step 1: make a table to find the combination of
numbers that multiply to “a × c” and add to “b”.
multiply to add to
step 2: use this combination of numbers to rewrite
the middle term of the trinomial as a polynomial with
four terms.
step 3: split the polynomial in half and find the gcf of
each set of binomials. factor out each gcf.
step 4: factor out the common binomial and use the
gcf of each set to write a second binomial.
the factored form of each trinomial.
2 5x² + 18x − 8
3 4x² − 12x − 7

Explanation:

Step1: Analyze the trinomial \(2x^2 + 15x + 18\)

For a trinomial \(ax^2+bx + c\) (here \(a = 2\), \(b = 15\), \(c = 18\)), we need two numbers that multiply to \(a\times c=2\times18 = 36\) and add to \(b = 15\). The factors of 36 are: (1,36), (2,18), (3,12), (4,9), (6,6). Among these, \(3 + 12=15\) and \(3\times12 = 36\).

Step2: Rewrite the middle term

Rewrite \(15x\) as \(3x+12x\), so the trinomial becomes \(2x^2+3x + 12x+18\).

Step3: Factor by grouping

Group the first two and last two terms: \((2x^2 + 3x)+(12x + 18)\). Factor out GCF from each group: \(x(2x + 3)+6(2x + 3)\).

Step4: Factor out the common binomial

Now, factor out \((2x + 3)\) from both terms: \((2x + 3)(x + 6)\).

Let's also factor the other trinomials:

For \(5x^2+18x - 8\)

• Step1: \(a = 5\), \(b = 18\), \(c=-8\). \(a\times c=5\times(-8)=-40\). Find two numbers that multiply to \(-40\) and add to \(18\). The numbers are \(20\) and \(-2\) (since \(20\times(-2)=-40\) and \(20+(-2)=18\)).
• Step2: Rewrite middle term: \(5x^2+20x-2x - 8\).
• Step3: Group: \((5x^2 + 20x)+(-2x - 8)\). Factor GCF: \(5x(x + 4)-2(x + 4)\).
• Step4: Factor out common binomial: \((5x - 2)(x + 4)\).

For \(4x^2-12x - 7\)

• Step1: \(a = 4\), \(b=-12\), \(c = -7\). \(a\times c=4\times(-7)=-28\). Find two numbers that multiply to \(-28\) and add to \(-12\). The numbers are \(-14\) and \(2\) (since \(-14\times2=-28\) and \(-14 + 2=-12\)).
• Step2: Rewrite middle term: \(4x^2-14x+2x - 7\).
• Step3: Group: \((4x^2 - 14x)+(2x - 7)\). Factor GCF: \(2x(2x - 7)+1(2x - 7)\).
• Step4: Factor out common binomial: \((2x - 7)(2x + 1)\).

Answer:

• \(2x^2 + 15x + 18=(2x + 3)(x + 6)\)
• \(5x^2+18x - 8=(5x - 2)(x + 4)\)
• \(4x^2-12x - 7=(2x - 7)(2x + 1)\)