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1.2. When the GCF is a binomial. |
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Prerequisite knowledge and skills: |
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Binomial |
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Factor |
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Factoring |
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Greatest Common Factor |
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Monomial |
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Term |
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Sometimes the GCF is a binomial. |
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Example 1.Factor
out the GCF: |
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Think of this expression as having two big terms: |
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The binomial |
is a factor in both terms. How many time is it in common to both? |
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The GCF is the binomial with the lower power: |
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Divide all the terms by the GCF |
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Multiply the two factors together: |
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Check 1: Are the tables of values equivalent? |
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Check 2: Are the graphical representations identical? |
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Example 2. Factor
out the GCF: |
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This expression, though longer, still has only two terms: |
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The binomials they common to both? |
are factors in both terms. How many time are
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The GCF consists of each common factor to the lower power: |
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The GCF is |
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Divide each term by the GCF and cancel fractions = to 1: |
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Multiply the factors together: |
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Check 1: Are the tables of values equivalent? |
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Check 2: Are the graphical representations identical? |
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We can extend this idea to include binomials taken to negative or fractional powers as we did in section 1.1 with monomial common factors. |
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Example 3. Factor out the term with
the lower power: |
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As before, we have two terms: |
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The
factor |
is common to both. How many times is it in common? |
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We take out the binomial with the lower power |
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Divide each term by this factor: |
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Multiply the factors together: |
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Check 1: Are the tables of values equivalent? |
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Check 2: Are the graphical representations identical? Where is the expression undefined? Where is the x-intercept? |
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Example 4.Factor
out the term with the lowest power: |
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The
factor |
is common to both terms. How many times is it in common? |
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We take out the binomial with the lower power: |
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Divide both terms by this factor: |
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Multiply the factors together: |
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Check 1: Are the tables of values equivalent? |
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Check 2: Are the graphical representations identical? Where is the expression undefined? Where is the x-intercept? |
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Example 5. Factor out the GCF: |
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The
factors 4, x,
and common to both terms? |
are common to both terms. How many times are they in |
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The GCF consists of each common factor to the lower power: |
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Divide each term by this GCF: |
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Multiply the factors together: |
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The checks are left for you. |
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