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How Do You Factor a Common Factor Out Of a Difference of Squares?

Factor 8x7-18xy2

Summary

  1. Factor out a greatest common factor of 2x
  2. The square root of 4, 2, is the coefficient of the first terms
  3. The square root of 9, 3, is the coefficient of the second terms
  4. In the first terms the variable is still x, and the exponent is half the degree, which is 3
  5. In the second terms the variable is still y, and the exponent is half the degree, which is 1

Notes

    1. Our polynomial is 8x7-18xy2
    1. 8 factors into 2•2•2
    2. 18 factors into 2•3•3
    1. x7 is the same as multiplying x by itself 7 times
    2. So each of our terms has at least one factor of x
    1. There is a factor of 2 and x in each of our terms
    2. This means we can factor them out of both terms
    1. To find the greatest common factor, just multiply together all the common factors
    2. In our case, the GCF is 2•x, or 2x
    1. To factor out the GCF, just divide it by each term in the polynomial
    1. 8x7/2x=4x6
    2. -18xy2/2x=-9y2
    1. Now that we've factored out the GCF, 2x, we need to factor what's left, 4x6-9y2
    1. The square root of 4 is 2
    2. The square root of 9 is 3
    1. The degree of x6 is 6
    2. The degree of y2 is 2
    1. Since both our coefficients are perfect squares and our degrees are even, we can use difference of squares
    1. We can use difference of squares to factor the polynomial
    1. 4x6 is the first term in our polynomial
    2. Its coefficient is 4
    3. Its variable is x
    4. Its degree is 6
    1. 9y2 is the second term in our polynomial
    2. Its coefficient is 9
    3. Its variable is y
    4. Its degree is 2
    1. When we factor the difference of two squares, it will always be an addition multiplied by a subtraction
    1. Remember, our coefficients were 4 and 9
    2. The coefficients of our new terms will be the square roots of 4 and 9
    1. 2 will be the coefficient of the first terms in our new binomials
    1. 3 will be the coefficient of the second terms in our new binomials
    1. Remember, our degrees were 6 and 2
    2. So half of 6 is 3 and half of 2 is 1
    1. We keep the same variable we had in the first term, x
    2. The 3 we got when we divided the degree in half is the new exponent
    1. We keep the same variable we had in the second term, y
    2. The 1 we got when we divided the degree in half is the new exponent
    1. We can do this by FOILing the binomials back together
    1. We can do this by FOILing
    2. Remember, FOIL is a method we use to multiply two binomials
    1. We need to multiply the first, outer, inner, and last terms of our binomials together
    2. Multiply the first terms, 2x3•2x3, to get 4x6
    3. Multiply the outer terms, 2x3•-3y, to get -6x3y
    4. Multiply the inner terms, 3y•2x3, to get 6x3y
    5. Multiply the last terms, 3y•-3y, to get -9y2
    1. -6x3y and 6x3y are like terms, because they have the same variables to the same degrees
    2. -6x3y+6x3y=0
    3. So the two terms cancel out, leaving us with 4x6-9y2
    1. Remember, we had a 2x out in front of our polynomial before we factored it
    1. The 2x is still going to be part of our factored polynomial