How do you factor the sum of terms as a product of the GCF and a sum?
It took me a while to interpret your question, and I came up with a polynomial that had a GCF, but it doesn't have to be a polynomial:
36x^2+45x+63 simplifies when we note that the GCF of the numbers 36, 45 and 63 is 9. That means 9 is the largest integer to divide into the numbers: 36=9*4, 45=9*5 and 63=9*7. This means we can rewrite the polynomial as the product of a GCF and a sum: 9(4x^2+5x+7). It utilises the distributive property of numbers.
Another example: abc+ace-ca^2. In this case the GCF is algebraic: ac, because all the terms contain ac, so we can factorise: ac(b+e-a). This is the product of the GCF ac and the sum b+e-a.
Another example: 4x^2+2xy+6x=2x(2x+y+3). This time the GCF is 2x because the GCF of the numbers is 2 and of the algebraic quantities is x, so we just combine the two GCFs to make 2x.
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