Question

Paula used synthetic division to divide the polynomial f(x) by x + 1, as shown:

What is the value of f(–1)?


7

–2


1

–1

Answer 1

Sorry I had to read up a sec :)
I sort of forgot synthetic lol

The Remainder Theorem tells us that when you perform this division,
the remainder will tell us the value of the function at that x value.

Do you know how to find the remainder in this division diagram?

Answer 2

think about what you compute using synthetic division:

$$(-2\cdot-1+-5)\cdot-1+4=-2\cdot(-1)^2-5\cdot(-1)+4=-2x^2-5x+4$$ evaluated at \(x=-1\)

Answer 3

the remainder (final) term in synthetic division gives you the value of the polynomial at that point; see the remainder theorem -- if we divide a polynomial \(P\) by a divisor \(D\) and get the quotient \(Q\) with remainder \(R\) we have:
$$P(x)=D(x)Q(x)+R(x)$$

if \(D\) has a zero at \(x=a\), then we see:
$$P(a)=D(a)Q(a)+R(a)\\\implies P(a)=R(a)\text{ since }D(a)=0$$

so if we divide by a monomial \(D(x)=x+1\), the remainder \(R(x)\) will tell us the value at the zero of \(D\) (which is \(-1\), since \(D(-1)=0\)). see:
$$P(-1)=D(-1)Q(-1)+R(-1)\\\implies P(-1)=R(-1)$$

so the remainder term givesu s what we want