SOLUTION: If a fourth degree polynomial is multiplied by a third degree polynomial, what is the degree of the product? Explain your reasoning and provise examples. Thank You

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Question 190673This question is from textbook
: If a fourth degree polynomial is multiplied by a third degree polynomial, what is the degree of the product? Explain your reasoning and provise examples.
Thank You This question is from textbook

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
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%5E4%2B3x and the third degree binomial x%5E3-10:



%28x%5E4%2B3x%29%28x%5E3-10%29 Start with the given expression.


Now let's FOIL the expression.


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


%28highlight%28x%5E4%29%2B3x%29%28highlight%28x%5E3%29-10%29 Multiply the First terms:%28x%5E4%29%2A%28x%5E3%29=x%5E7.


%28highlight%28x%5E4%29%2B3x%29%28x%5E3%2Bhighlight%28-10%29%29 Multiply the Outer terms:%28x%5E4%29%2A%28-10%29=-10%2Ax%5E4.


%28x%5E4%2Bhighlight%283x%29%29%28highlight%28x%5E3%29-10%29 Multiply the Inner terms:%283%2Ax%29%2A%28x%5E3%29=3%2Ax%5E4.


%28x%5E4%2Bhighlight%283x%29%29%28x%5E3%2Bhighlight%28-10%29%29 Multiply the Last terms:%283%2Ax%29%2A%28-10%29=-30%2Ax.


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So we have the terms: x%5E7, -10%2Ax%5E4, 3%2Ax%5E4, -30%2Ax


x%5E7-10%2Ax%5E4%2B3%2Ax%5E4-30%2Ax Now add every term listed above to make a single expression.


x%5E7-7%2Ax%5E4-30%2Ax Now combine like terms.


So %28x%5E4%2B3x%29%28x%5E3-10%29 FOILs to x%5E7-7%2Ax%5E4-30%2Ax.


In other words, %28x%5E4%2B3x%29%28x%5E3-10%29=x%5E7-7x%5E4-30%2Ax.


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