When a certain polynomial is divided by x 3, its quotient is x2 5x 7 and its remainder is 5. What is the polynomial? (hint: use the division statement!)
The division statement can be written as:
Polynomial = Divisor * Quotient + Remainder
Since the divisor is (x - 3), the polynomial can be expressed as:
Polynomial = (x - 3) * (x^2 - 5x - 7) + 5
We can now multiply the divisor and quotient:
Polynomial = x * (x^2 - 5x - 7) - 3 * (x^2 - 5x - 7) + 5
Simplifying, we get:
Polynomial = x^3 - 5x^2 - 7x - 3x^2 + 15x + 21 + 5
Combining like terms, the polynomial is:
Polynomial = x^3 - 8x^2 + 8x + 26
To find the polynomial, we can use the division statement:
Dividend = Divisor × Quotient + Remainder
Let's assign variables to the dividend, divisor, quotient, and remainder.
Dividend = P(x)
Divisor = (x - 3)
Quotient = (x^2 - 5x - 7)
Remainder = 5
So, the division statement becomes:
P(x) = (x - 3) × (x^2 - 5x - 7) + 5
Now, we can expand the expression on the right side:
P(x) = (x^3 - 5x^2 - 7x) - 3(x^2 - 5x - 7) + 5
Simplifying further:
P(x) = x^3 - 5x^2 - 7x - 3x^2 + 15x + 21 + 5
Combining like terms:
P(x) = x^3 - 8x^2 + 8x + 26
Therefore, the polynomial is P(x) = x^3 - 8x^2 + 8x + 26.
To find the polynomial, we will use the division statement. We are given that when the polynomial is divided by (x - 3), the quotient is (x^2 - 5x - 7) and the remainder is 5.
The division statement can be written as:
Polynomial = Divisor × Quotient + Remainder
In this case, the polynomial is represented by an unknown, which we can call it P(x). The divisor is (x - 3), the quotient is (x^2 - 5x - 7), and the remainder is 5. So, the division statement becomes:
P(x) = (x - 3) × (x^2 - 5x - 7) + 5
Now, let's multiply the divisor by the quotient:
P(x) = (x - 3) × (x^2 - 5x - 7) + 5
P(x) = x(x^2 - 5x - 7) - 3(x^2 - 5x - 7) + 5
P(x) = x^3 - 5x^2 - 7x - 3x^2 + 15x + 21 + 5
P(x) = x^3 - 5x^2 - 3x^2 - 7x + 15x + 21 + 5
P(x) = x^3 - 8x^2 + 8x + 26
Therefore, the polynomial is x^3 - 8x^2 + 8x + 26.