Mathematics, 08.11.2019 05:31, pearfam98

Given a polynomial and one of its factors, find the remaining factors of the polynomial. some of the factors may not be binomials. 16x3 – 48x2 – 49x + 147; x – 3

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The correct answer was given: dxnimxriee

remaining factors: (x +1)(x² +2x +3)

Step-by-step explanation:

Synthetic division can be used to divide the given polynomial by the given factor. The result is ...

x^3+3x^2+5x+3

We observe there are no changes in sign here, so the factor (x-1) represents the only positive real root. When we change the signs of the odd-degree terms, we now have 3 sign changes, indicating 3 or 1 negative real roots. We also observe that the sign-altered coefficients sum to zero, meaning that x=-1 is a real root of the polynomial.

At this point, we know a second factor is (x+1). Again using synthetic division, we find the remaining factor is a quadratic with no real roots:

x^2 +2x +3

The remaining factors (after (x-1)) with real coefficients are ...

(x +1)(x² +2x +3)

_____

The quadratic can be factored in the complex numbers as ...

(x +1 -i√2)(x +1 +i√2)

_____

Comment on sign changes

The discussion of sign changes refers to Descartes' rule of signs. It says the number of positive real roots is equal to the number of changes of sign of the coefficients when the polynomial is written in standard form. That number may be reduced by multiples of 2 if some of the roots are complex.

Hence the 3 sign changes we observe in -x^3 +3x^2 -5x +3 means there are either 3 or 1 negative real roots. (If f(x) has negative real roots, f(-x) will have positive real roots.)

Comment on the graph

For higher-degree polynomials, I like to see what information a graphing calculator can provide about roots. Here, it tells us the real roots are -1 and +1 and the other two roots are complex. Factoring out the factors related to the real roots gives the quadratic (x+1)^2 +2 = x^2 +2x +3, as above. The vertex of the quadratic gives useful information about the complex roots.

$Given a polynomial and one of its factors, find the remaining factors of the polynomial. [tex]x^{4}+$
$Given a polynomial and one of its factors, find the remaining factors of the polynomial. [tex]x^{4}+$
$Given a polynomial and one of its factors, find the remaining factors of the polynomial. [tex]x^{4}+$

The correct answer was given: chloerodgers56

The other factors are (4x + 7) and (4x - 7).

Step-by-step explanation:

One of the factor of the polynomial $16x^{3} - 48x^{2} - 49x + 147$ is given to be (x - 3). We have to find the other factors of this polynomial.

Now, $16x^{3} - 48x^{2} - 49x + 147$

= $16x^{2} (x - 3) - 49(x - 3)$

= $(x - 3)( 16x^{2} - 49)$

= $(x - 3)[(4x)^{2} - 7^{2}]$ {Since we know the formula, $(a^{2}-b^{2} ) = (a + b)(a - b)$ }

= (x - 3)(4x + 7)(4x - 7)

Therefore, the other factors are (4x + 7) and (4x - 7). (Answer)

The correct answer was given: mrus

x=1

Step-by-step explanation:

The correct answer was given: kaylaciamp65

Find attached solution. Cheers.

The second quadratic function is not factorisable, so we leave it that way.

Cheers.

The correct answer was given: cbaublitz

The remaining factors is x+1 and x+4
Given a polynomial and one of its factors, find the remaining factors of the polynomial. x3 + 3x2 –

The correct answer was given: andrew194

This is easy to do because you can factor
example
6/2=3 because 6=2*3 so 6/2=3/1 times 2/2 or 3

2x^3+17x^2+23x-42 can be factored out to equal
(x-1)(x+6)(2x+7)

so [(x-1)(x+6)(2x+7)]/(2x+7)=[(x-1)(x+6)] times (2x+7)/(2x+7)=(x-1)(x+6)=x^2+5x-6

the answer is (x-1)(x+6) or x^2+5x-6

Given a polynomial and one of its factors, find the remaining factors of the polynomial. some factor

Given a polynomial and one of its factors, find the remaining factors of the polynomial. some factor

The correct answer was given: solizpaco7124

This is easy to do because you can factor
example
6/2=3 because 6=2*3 so 6/2=3/1 times 2/2 or 3

2x^3+17x^2+23x-42 can be factored out to equal
(x-1)(x+6)(2x+7)

so [(x-1)(x+6)(2x+7)]/(2x+7)=[(x-1)(x+6)] times (2x+7)/(2x+7)=(x-1)(x+6)=x^2+5x-6

the answer is (x-1)(x+6) or x^2+5x-6

given a polynomial and one of its factors, find the remaining factors of the polynomial. some factor