Question

7. By using the factor theorem, show that (x + 2) is
a factor of the polynomial 6x3 + 19x2 + 16x + 4
and then factorise 6x3 + 19x2 + 16x + 4.​

Answer 1

Answer:

(6x3-19x2+16x-4)/(x-2)

Final result :

(2x - 1) • (3x - 2)

Reformatting the input :

Changes made to your input should not affect the solution:

(1): "x2" was replaced by "x^2". 1 more similar replacement(s).

Step by step solution :

Step 1 :

Equation at the end of step 1 :

Step 2 :

Equation at the end of step 2 :

Step 3 :

6x3 - 19x2 + 16x - 4

Simplify ————————————————————

x - 2

Checking for a perfect cube :

3.1 6x3 - 19x2 + 16x - 4 is not a perfect cube

Trying to factor by pulling out :

3.2 Factoring: 6x3 - 19x2 + 16x - 4

Thoughtfully split the expression at hand into groups, each group having two terms :

Group 1: 16x - 4

Group 2: 6x3 - 19x2

Pull out from each group separately :

Group 1: (4x - 1) • (4)

Group 2: (6x - 19) • (x2)

Bad news !! Factoring by pulling out fails :

The groups have no common factor and can not be added up to form a multiplication.

Polynomial Roots Calculator :

3.3 Find roots (zeroes) of : F(x) = 6x3 - 19x2 + 16x - 4

Polynomial Roots Calculator is a set of methods aimed at finding values of x for which F(x)=0

Rational Roots Test is one of the above mentioned tools. It would only find Rational Roots that is numbers x which can be expressed as the quotient of two integers

The Rational Root Theorem states that if a polynomial zeroes for a rational number P/Q then P is a factor of the Trailing Constant and Q is a factor of the Leading Coefficient

In this case, the Leading Coefficient is 6 and the Trailing Constant is -4.

The factor(s) are:

of the Leading Coefficient : 1,2 ,3 ,6

of the Trailing Constant : 1 ,2 ,4

Let us test ....

P Q P/Q F(P/Q) Divisor

-1 1 -1.00 -45.00

-1 2 -0.50 -17.50

-1 3 -0.33 -11.67

-1 6 -0.17 -7.22

-2 1 -2.00 -160.00

-2 3 -0.67 -24.89

-4 1 -4.00 -756.00

-4 3 -1.33 -73.33

1 1 1.00 -1.00

1 2 0.50 0.00 2x - 1

1 3 0.33 -0.56

1 6 0.17 -1.83

2 1 2.00 0.00 x - 2

2 3 0.67 0.00 3x - 2

4 1 4.00 140.00

4 3 1.33 -2.22

The Factor Theorem states that if P/Q is root of a polynomial then this polynomial can be divided by q*x-p Note that q and p originate from P/Q reduced to its lowest terms

In our case this means that

6x3 - 19x2 + 16x - 4

can be divided by 3 different polynomials,including by 3x - 2

Polynomial Long Division :

3.4 Polynomial Long Division

Dividing : 6x3 - 19x2 + 16x - 4

("Dividend")

By : 3x - 2 ("Divisor")

dividend 6x3 - 19x2 + 16x - 4

- divisor * 2x2 6x3 - 4x2

remainder - 15x2 + 16x - 4

- divisor * -5x1 - 15x2 + 10x

remainder 6x - 4

- divisor * 2x0 6x - 4

remainder 0

Quotient : 2x2-5x+2 Remainder: 0

Trying to factor by splitting the middle term

3.5 Factoring 2x2-5x+2

The first term is, 2x2 its coefficient is 2 .

The middle term is, -5x its coefficient is -5 .

The last term, "the constant", is +2

Step-1 : Multiply the coefficient of the first term by the constant 2 • 2 = 4

Step-2 : Find two factors of 4 whose sum equals the coefficient of the middle term, which is -5 .

-4 + -1 = -5 That's it

Step-3 : Rewrite the polynomial splitting the middle term using the two factors found in step 2 above, -4 and -1

2x2 - 4x - 1x - 2

Step-4 : Add up the first 2 terms, pulling out like factors :

2x • (x-2)

Add up the last 2 terms, pulling out common factors :

1 • (x-2)

Step-5 : Add up the four terms of step 4 :

(2x-1) • (x-2)

Which is the desired factorization

Canceling Out :

3.6 Cancel out (x-2) which appears on both sides of the fraction line.

Final result :

(2x - 1) • (3x - 2)

Step-by-step explanation:

hope hope it helpful to us

Answer 2

$p(x) = 6 {x}^{3} + 19 {x}^{2} + 16x + 4 \\ g(x) = x + 2$

therefore,x= –2

$p( - 2) = 6 ({ - 2})^{3} + 19 ({ - 2})^{2} + 16( - 2) + 4 \\ = - 48 + 76 - 36 + 4 \\ = - 80 + 80 \\ = 0$

x=(-2) is zero of p(x)

therefore, x+2 is a factor of p(x)

$6x^3 + 19x ^2 + 16x + 4 \div x + 2 \\ = 6x^{2} + 7x + 2$

$q(x) = 6 {x}^{2} + 7x + 2 \\ r(x) = 0$

$p(x) = g(x) \times q(x) + r(x) \\ p(x) = (x + 2)(6 {x}^{2} + 7x + 2) + 0 \\ p(x) = (x + 2)(6 {x}^{2} + 4x + 3x + 2) \\ p(x) =( x + 2)(2x(3x + 2) + 1(3x + 2)) \\ p(x) = (x + 2)(2x + 1)(3x + 2)$