We can use the Polynomial Remainder Theorem. It states that if we divide a polynomial P(x) by a binomial in the form (x - a), then our remainder will be P(a).

We are dividing:

$(x^2+5x+7)\div(x+k)$

So, a polynomial by a binomial factor.

Our factor is (x + k) or (x - (-k)). Using the form (x - a), our a = -k.

We want our remainder to be 3. So, P(a)=P(-k)=3.

Therefore:

$(-k)^2+5(-k)+7=3$

Simplify:

$k^2-5k+7=3$

Solve for k. Subtract 3 from both sides:

$k^2-5k+4=0$

Factor:

$(k-1)(k-4)=0$

Zero Product Property:

$k-1=0\text{ or } k-4=0$

Solve:

$k=1\text{ or } k=4$

So, either of the two expressions:

$(x^2+5x+7)\div(x+1)\text{ or } (x^2+5x+7)\div(x+4)$

Will yield 3 as the remainder.

5 0

3 years ago

Michael saves $423 dollars a month for college.

alexandr1967 [171]

Answer:

(a). $20,000

(b). The estimate will be lower than the actual amount.

Step-by-step explanation:

We have been given that Michael saves $423 dollars a month for college.

(a). We know that 1 year equals 12 months.

4 years = 4*12 months = 48 months.

$\text{Actual amount saved}=\$423\times 48$

Since we are asked to find the estimated amount of money Michael will save in 4 years, so we will estimate both quantities as:

$423\approx 400\\\\48\approx 50$

$\text{Actual amount saved}=\$400\times 50$

$\text{Actual amount saved}=\$20,000$

Therefore, Michael will save approximately $20,000 in 4 years.

(b).

The estimate will be lower than the actual amount as we rounded $423 down $23 to nearest hundred that is $400 and rounded 48 up 2 to nearest ten that is 40.

Therefore, the estimate will be lower than the actual amount.

5 0

3 years ago