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Multiplying Polynomials

Sometimes algebra problems will ask you to combine polynomials with multiplication.  These problems can look intimidating, but they are very similar to multiplying binomials (remember FOIL?).  The secret to these problems is being methodical and making sure that every term in each polynomial is multiplied by every term in each of the other polynomials. 

Remember that:

  • When you multiply numbers together (even if there are variables attached) you multiply the numbers.
  • When you multiply the same variable together, you change the exponent, following the "multiplying exponents rules" (e.g., $a \times a = a^2$ and $b^2\times b^2=b^4$).
  • When you multiply different variables together, you just stick them together (e.g., $f\times g = fg$).



Just like when you add and subtract polynomials, the best way to do these kinds of problems meticulously.  In this case, rather than looking for like terms, start with the first term in the first polynomial and multply it times each term in the second, then take the next term in the first polynomial and multiply it times each term in the second, and so on, until you have multiplied each term.  Then combine like terms! 

$$\eqalign{(2x^2+6x-9)(4x^2+5x+10)&=\\\text{First term in first polynomial times each term in the next polynomial:}\\(\boxed{2x^2}+6x-9)(4x^2+5x+10)&=8x^4 +10x^3+20x^2\\\text{Second term in first polynomial times each term in next polynomial:}\\(\cancel{2x^2}+\boxed{6x}-9)(4x^2+5x+10)&=8x^4+10x^3+20x^2\mathbf{+24x^3+30x^2+60x}\\\text{Last term in first polynomial times each term in second polynomial:}\\(\cancel{2x^2+6x}\boxed{-9})(4x^2+5x+10)&=8x^4+10x^3+20x^2+24x^3+30x^2+60x+\mathbf{-36x^2-45x-90}\\\text{Combine like terms:}\\8x^4+10x^3+20x^2+24x^3+30x^2+60x-36x^2-45x-90&=\mathbf{8x^4+34x^3+14x^2+15x-90}}$$

As you can see, you have to be very careful with these problems.  It's very easy to drop a term or confuse a negative.  Plus, there's a ton of combining like terms that needs to be done at the end.  But work them carefully and you'll see they are not hard!

Practice Problems:

  • Multiplying Polynomials

    1. $(4x^2+7x)(6x^2-9x-10)$

    2. $(2x^3+x)(x^2-x-1)$

    3. $(-x^2-3y+7x)(2x^2-5y-3)$

    4. $(x^4+3x-4)(x^2-y-9)$

    5. $(4x+7)(-2x^3-9x^2+5)$

    6. $(3x^3+7x^2+8x+4)(3x+4)$

    7. $(x^2+7z-8)(-x^2-2x-12)$

    8. $(5x^2-5x)(-3x^2-2x-5)$

    9. $(-3x^2-7y^2)(x^2-y-10x)$

    10. $(4x^2+x+1)(4x^2+x+1)$

    11. $(x^5+7x^3)(x^2+1x+3)$

    12. $(-x-y-z)(x+y+z)$


Common Core Grade Level/Subject

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