Question

1.
POLYNOMIALS
Find ax" + bx' + c to be polynomial p & q are :
a) Rational Numbers​

Answer 1

n the same way as you could factor trinomials on the form of

x2+bx+c

You can factor polynomials on the form of

ax2+bx+c

If a is positive then you just proceed in the same way as you did previously except now

ax2+bx+c=(x+m)(ax+n)

wherec=mn,ac=pqandb=p+q=am+n

Example

3x2−2x−8

We can see that c (-8) is negative which means that m and n does not have the same sign. We now want to find m and n and we know that the product of m and n is -8 and the sum of m and n multiplied by a (3) is b (-2) which means that we're looking for two factors of -24 whose sum is -2 and we also know that one of them is positive and of them is negative.

Factorsof−24−1,241,−24−2,122,−12−3,83,−8−4,64,−6Sumoffactors23−2310−105−52−2

This means that:

3x2−2x−8=

=3x2+(4−6)x−8=

=3x2+4x−6x−8

We can then group those terms that have a common monomial factor. The first two terms have x together and the second two -2 and then factor the two groups.

=(3x2+4x)+(−6x−8)=

=x(3x+4)−2(3x+4)

Notice that both remaining parenthesis are the same. This means that we can rewrite this using the distributive propertyit as:

=(x−2)(3x+4)=3x2−2x−8

This method is called factor by grouping.

A polynomial is said to be factored completely if the polynomial is written as a product of unfactorable polynomials with integer coefficients.