## anonymous one year ago 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

1. anonymous

2. zepdrix

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?

3. anonymous

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 $$x=-1$$

4. anonymous

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