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marta [7]
3 years ago
15

Use the Remainder Theorem to completely factor p(x) = x ^3 + 7x ^2 + 11x + 5.

Mathematics
2 answers:
notka56 [123]3 years ago
5 0

Answer: (x+1)(x+1)(x+5)

Step-by-step explanation:

let f(x) = x^3 + 7x^2 + 11x +5

let x= -1

f(-1)= (-1)^3 + 7(-1)^2 + 11(-1) + 5

= -1 +7-11+5

=0

Therefore x + 1 is a factor

Then we proceed to divide

x^3 + 7x^2 + 11x +5 by x+1

_x^2+6x+5____

x+1√x^3 + 7x^2 + 11x + 5

-(x^3 + x^2)

____________________

6x^2 + 11x

- (6x^2 + 6x)

_______________________

5x +5

- (5x + 5)

_______________________

0

so after division,

x^2+6x+5 is also another factor, but we can further break it down into 2 factors.

let f(x) = x^2 + 6x + 5

let x= -1

f(-1)= (-1)^2 + 6(-1) + 5

= 1 - 6 + 5

= 0

Therefore (x+1) is also another factor,

we can use x+ 1 to divide

x^2 + 6x + 5 to get the next factor

_____x+5_____

x+1√x^2+6x +5

- (x^2+6x)

_____________

5x + 5

- (5x+5)

______________

0

Therefore x + 5 is another factors

(x+1)(x+1)(x+5) are the factors of the polynomial.

tigry1 [53]3 years ago
4 0

In order to use the remainder theorem, you need to have some idea what to divide by. The rational root theorem tells you rational roots will be from the list derived from the factors of the constant term, {±1, ±5}. When we compare coefficients of odd power terms to those of even power terms, we find their sums are equal, which means -1 is a root and (x +1) is a factor.

Dividing that from the cubic, we get a quotient of x² +6x +5 (and a remainder of zero). We recognize that 6 is the sum of the factors 1 and 5 of the constant term 5, so the factorization is

... = (x +1)(x +1)(x +5)

... = (x +1)²(x +5)

_____

The product of factors (x +a)(x +b) will be x² + (a+b)x + ab. That is, the factorization can be found by looking for factors of the constant term (ab) that add to give the coefficient of the linear term (a+b). The numbers found can be put directly into the binomial factors to make (x+a)(x+b).

When we have 1·5 = 5 and 1+5 = 6, we know the factorization of x²+6x+5 is (x+1)(x+5).

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132

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Check:

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Miguel biked 14 km in the same amount of time that linda hiked 10 km. Miguel's rate was 4 km/h more than linda's. How fast did e
matrenka [14]

The first thing we must do is define a variable.

We have then:

x: Linda's speed.

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We have then:

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\frac{10}{x} = \frac{14}{x + 4}

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10 (x + 4) = 14x

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3 years ago
Waqas wishes to distribute 128 bananas and also 176 apples equally among a certain number of children. find the highest number o
atroni [7]
Answer:
The maximum number of children = 16 

Explanation:
The greatest number of students who can get the fruit in this way (equally) will be equal to the highest common factor between the number of bananas and the number of apples.

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This means that the maximum number of children to get both equally is 16

Hope this helps :)
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