Let p(x) = x^4 - 7x + 14 and q(x) = x^2 - 5
a) What is the degree of the polynomial p(x) + q(x)
b) What is the degree of the polynomial p(x) - q(x)
c) What is the degree of the polynomial p(x) * q(x)
d) In general, if p(x) and q(x) are polynomials such that p(x) has a degree m. q(x) has a degree n, and m>x, what are the degrees of p(x) + q(x), p(x) - q(x) and p(x) * q(x).
I seriously don't get these lol, my teacher just jump away. Plz help!
a) What is the degree of the polynomial p(x) + q(x)
b) What is the degree of the polynomial p(x) - q(x)
c) What is the degree of the polynomial p(x) * q(x)
d) In general, if p(x) and q(x) are polynomials such that p(x) has a degree m. q(x) has a degree n, and m>x, what are the degrees of p(x) + q(x), p(x) - q(x) and p(x) * q(x).
I seriously don't get these lol, my teacher just jump away. Plz help!
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The "degree" of a polynomial is the exponent of the highest-degree variable.
For example, p(x) is a "degree-4" polynomial (also called a "quartic" by us nerds)
When you add two polynomials, you add variables of equal-degree together.
Therefore, adding p and q will never change the x^4 term, because there is nothing in q that gets added (or subtracted) at that level.
When multiplying two polynomials, you "distribute" the terms of one polynomial into the other polynomial:
p(x)*q(x) = (x^4 - 7x + 14)(x^2 - 5)
= x^4(x^2 - 5) - 7x(x^2 - 5) + 14(x^2 - 5)
= (x^4)(x^2) + (x^4)(-5) + (-7x)(x^2) + (-7x)(-5) + (14)(x^2) + (14)(-5)
= x^6 + ...
If you quickly look at all the other terms, you will see that none will be bigger (in power) than x^6
Question d simply asks you to create a general rule.
There is a typo in your question, it is "m>n" not m>x.
p(x) = x^m + k x^(m-1) + ...w x^n + ...
q(x) = x^n + r x^(n-1) + ...
where n is smaller than m
then nothing in q will ever change the x^m in p
Therefore, a + q and p - q will always have x^m as the highest term.
When you multiply p * q, the highest-degree term of the product will always be (x^m)(x^n) = x^(m+n)
For example, p(x) is a "degree-4" polynomial (also called a "quartic" by us nerds)
When you add two polynomials, you add variables of equal-degree together.
Therefore, adding p and q will never change the x^4 term, because there is nothing in q that gets added (or subtracted) at that level.
When multiplying two polynomials, you "distribute" the terms of one polynomial into the other polynomial:
p(x)*q(x) = (x^4 - 7x + 14)(x^2 - 5)
= x^4(x^2 - 5) - 7x(x^2 - 5) + 14(x^2 - 5)
= (x^4)(x^2) + (x^4)(-5) + (-7x)(x^2) + (-7x)(-5) + (14)(x^2) + (14)(-5)
= x^6 + ...
If you quickly look at all the other terms, you will see that none will be bigger (in power) than x^6
Question d simply asks you to create a general rule.
There is a typo in your question, it is "m>n" not m>x.
p(x) = x^m + k x^(m-1) + ...w x^n + ...
q(x) = x^n + r x^(n-1) + ...
where n is smaller than m
then nothing in q will ever change the x^m in p
Therefore, a + q and p - q will always have x^m as the highest term.
When you multiply p * q, the highest-degree term of the product will always be (x^m)(x^n) = x^(m+n)