Question 190673
The product of a fourth degree polynomial and a third degree polynomial is a 7th degree polynomial (just add the two degrees). Why is this the case? Remember, when you multiply variables with common bases, you add the exponents. Since the degree of a polynomial is just the largest exponent, you're really just adding the degrees when you multiply




Example: Let's multiply the fourth degree binomial {{{x^4+3x}}} and the third degree binomial {{{x^3-10}}}:




{{{(x^4+3x)(x^3-10)}}} Start with the given expression.



Now let's FOIL the expression.



Remember, when you FOIL an expression, you follow this procedure:



{{{(highlight(x^4)+3x)(highlight(x^3)-10)}}} Multiply the <font color="red">F</font>irst terms:{{{(x^4)*(x^3)=x^7}}}.



{{{(highlight(x^4)+3x)(x^3+highlight(-10))}}} Multiply the <font color="red">O</font>uter terms:{{{(x^4)*(-10)=-10*x^4}}}.



{{{(x^4+highlight(3x))(highlight(x^3)-10)}}} Multiply the <font color="red">I</font>nner terms:{{{(3*x)*(x^3)=3*x^4}}}.



{{{(x^4+highlight(3x))(x^3+highlight(-10))}}} Multiply the <font color="red">L</font>ast terms:{{{(3*x)*(-10)=-30*x}}}.



---------------------------------------------------

So we have the terms: {{{x^7}}}, {{{-10*x^4}}}, {{{3*x^4}}}, {{{-30*x}}} 



{{{x^7-10*x^4+3*x^4-30*x}}} Now add every term listed above to make a single expression.



{{{x^7-7*x^4-30*x}}} Now combine like terms.



So {{{(x^4+3x)(x^3-10)}}} FOILs to {{{x^7-7*x^4-30*x}}}.



In other words, {{{(x^4+3x)(x^3-10)=x^7-7x^4-30*x}}}.



Notice how the degree of the final answer is 7. So this confirms the original claim.