Questions With Solutions

How to factor a polynomial by grouping? Questions with detailed solutions and explanations are presented.

Factoring a polynomial by grouping is explained using several questions with their solutions.

__Question 1__

Factor completely the polynomial *4 x ^{ 2} + 4 x + 3 x + 3*

Note that all four terms in the given polynomial have no common factor.

However by grouping the first two term, we can factor

We now group the last two term and factor

Rewrite the given polynomial with the grouped terms in factored form.

Note that

__Question 2__

Factor the polynomial *2 x ^{ 2} - 4 x + 3 x y - 6 y*

There is no common factor to all 4 terms in the given polynomial.

Group the first two terms and factor

Group the last two terms and factor

Rewrite the given polynomial as follows

and factor out the common factor

__Question 3__

Factor the polynomial *x y - x - 2 y + 2*

__Solution to Question 3__

The terms in the given polynomial have no common factor.

The first two terms can be grouped and factored as follows: *x* out:

*x y - x = x ( y - 1)*

The last two terms can factored as: *2* out:

*- 2 y + 2 = 2( - y + 1) = - 2(y - 1)*

Rewrite the given polynomial in factored form as follows:

*x y - x - 2 y + 2 = x (y - 1) - 2 (y - 1) *

Factor out the common factor *(y - 1)* to factor completely

*x y - x - 2 y + 2 = (y - 1) - 2 (y - 1) = (y - 1)(x - 2)*

__Question 4__

Factor completely the polynomial 3* x ^{ 2} + 4 x + 1*

There is no common factor to the terms in the given polynomial. One way is to rewrite the polynomial with 4 terms that may be factored by grouping.

We use the identity

We group the first two terms and factor

Rewrite the given polynomial with the grouped terms in factored form.

Note that

b)

c)

d)

e)

f)

Solution to Question a)

We first find a common factor in * 2 x ^{ 2} - 4* and factor it as follows:

We next find a common factor in the

Use the common factor

Solution to Question b)

Find a common factor in

We next find a common factor in the

Use the common factor

The graph of the given polynomial in b) above

Solution to Question c)

Find a common factor in

We next find a common factor in the

Use the common factor

Solution to Question d)

The given polynomial has three terms with no common factor. One way to factor is to rewrite it replacing

We can now factor

We next factor

Use the common factor

Solution to Question e)

Note that

Rewrite by grouping terms as follows

The terms in

Solution to Question f)

Note that there are 5 terms in the given polynomial with common factor to all of them. Rewrite the polynomial replacing

We shall now factor the equivalent polynomial

We now group the last three terms and factor as follows

The two groups have the common factor

Factor Polynomials

Factoring of Special Polynomials

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