Question

fill in the missing values below one at a time to find the quotient when -3x³ - 14x² - 24x - 15 is divided by -3x - 5. -3x -3x³ -5 answer attempt 1 out of 2 you must answer all questions above in order to submit.

Explanation:

Step1: Find the first term of the quotient

We divide the leading term of the dividend \(-3x^3\) by the leading term of the divisor \(-3x\). So, \(\frac{-3x^3}{-3x}=x^2\). This is the first term of the quotient (top row, first box).

Step2: Multiply the divisor by \(x^2\)

Now, we multiply \(-3x - 5\) by \(x^2\).
- Multiply \(-3x\) by \(x^2\): \(-3x\times x^2=-3x^3\) (already given in the table).
- Multiply \(-5\) by \(x^2\): \(-5\times x^2 = -5x^2\) (bottom row, first box).

Step3: Find the second term of the quotient

Next, we look at the second term of the dividend, which is \(-14x^2\). We already have \(-3x^3\) accounted for, and the term from the multiplication of \(-5\) and \(x^2\) is \(-5x^2\). Wait, actually, we need to find the next term of the quotient. Let's consider the current sum of the middle column. The middle column has \(-3x^3\) (from \(-3x\times x^2\)) and we need to get to \(-14x^2\) when we add the terms from \(-3x\) and \(-5\) multiplications. Wait, maybe a better way: after the first term \(x^2\), we now have the remaining polynomial: \((-3x^3 - 14x^2 - 24x - 15)-(-3x^3 - 5x^2)=-9x^2 - 24x - 15\). Now, we divide the leading term of this remaining polynomial \(-9x^2\) by the leading term of the divisor \(-3x\). So, \(\frac{-9x^2}{-3x}=3x\). Wait, that can't be. Wait, no, let's check again. Wait, the divisor is \(-3x - 5\), and the dividend is \(-3x^3 -14x^2 -24x -15\). Let's do polynomial long division properly.

Wait, maybe the table is a rectangle method (area model) for polynomial division. The top row is the quotient \(ax^2 + bx + c\), the left column is the divisor \(-3x - 5\), and the middle is the product (dividend). So, the product of the quotient and divisor is the dividend. So, \((ax^2 + bx + c)(-3x - 5)=-3x^3 -14x^2 -24x -15\). We know that \(ax^2\times(-3x)=-3x^3\), so \(a = 1\) (since \(1\times(-3x)=-3x\) for the \(x^3\) term). So the first term of the quotient is \(x^2\) (top row, first box). Then, \(ax^2\times(-5)=-5ax^2=-5x^2\) (bottom row, first box). Now, the middle column (second column) comes from \(bx\times(-3x)\) and \(bx\times(-5)\)? Wait, no, the table has three columns (since the dividend is cubic, quotient is quadratic, so 3 terms: \(x^2\), \(x\), constant). So the top row is \(x^2\), \(bx\), \(c\) (constant). The left column is \(-3x\) and \(-5\). So the product table is:

| | \(x^2\) | \(bx\) | \(c\) |
|----------|-----------|-----------|-----------|
| \(-3x\) | \(-3x^3\) | \(-3bx^2\)| \(-3cx\) |
| \(-5\) | \(-5x^2\) | \(-5bx\) | \(-5c\) |

Now, the sum of the middle column ( \(x^2\) terms) is \(-3bx^2 -5x^2=-14x^2\) (from the dividend). So, \(-3bx^2 -5x^2=-14x^2\). Factor out \(x^2\): \((-3b -5)x^2=-14x^2\). So, \(-3b -5=-14\). Solving for \(b\): \(-3b=-14 + 5=-9\), so \(b = 3\). So the second term of the quotient is \(3x\) (top row, second box).

Step4: Fill in the middle column

Now, with \(b = 3\):
- Multiply \(-3x\) by \(3x\): \(-3x\times3x=-9x^2\) (middle row, second box).
- Multiply \(-5\) by \(3x\): \(-5\times3x=-15x\) (bottom row, second box).

Step5: Find the third term of the quotient

Now, the remaining polynomial after the first two terms: We have the \(x\) terms. The dividend has \(-24x\), and from the middle row ( \(-3x\times3x=-9x^2\) and \(-3x\times x^2=-3x^3\)) and bottom row ( \(-5\times x^2=-5x^2\) and \(-5\times3x=-15x\)), we now look at the \(x\) terms. The sum of the \(x\) terms from the product should be \(-24x\). The \(x\) term from \(-3x\times c\) and \(-5\times3x\) should add up to \(-24x\). Wait, the \(x\) term from \(-3x\times c\) is \(-3cx\) and from \(-5\times3x\) is \(-15x\). So, \(-3cx -15x=-24x\). Factor out \(x\): \((-3c -15)x=-24x\). So, \(-3c -15=-24\). Solving for \(c\): \(-3c=-24 + 15=-9\), so \(c = 3\). So the third term of the quotient is \(3\) (top row, third box).

Step6: Fill in the last column

Now, with \(c = 3\):
- Multiply \(-3x\) by \(3\): \(-3x\times3=-9x\) (middle row, third box). Wait, no, wait: the middle row is \(-3x\) times the quotient terms. So \(-3x\times3=-9x\) (middle row, third box).
- Multiply \(-5\) by \(3\): \(-5\times3=-15\) (bottom row, third box).

Now, let's verify the product:
- Top row: \(x^2\), \(3x\), \(3\)
- Left column: \(-3x\), \(-5\)
- Product:
- \(-3x\times x^2=-3x^3\)
- \(-3x\times3x=-9x^2\)
- \(-3x\times3=-9x\)
- \(-5\times x^2=-5x^2\)
- \(-5\times3x=-15x\)
- \(-5\times3=-15\)
- Now, sum the terms:
- \(x^3\) term: \(-3x^3\)
- \(x^2\) terms: \(-9x^2 -5x^2=-14x^2\)
- \(x\) terms: \(-9x -15x=-24x\)
- constant term: \(-15\)
- Which matches the dividend \(-3x^3 -14x^2 -24x -15\).

So the table should be filled as follows:

Top row (quotient): \(x^2\), \(3x\), \(3\)

Middle row ( \(-3x\) times quotient): \(-3x^3\), \(-9x^2\), \(-9x\)

Bottom row ( \(-5\) times quotient): \(-5x^2\), \(-15x\), \(-15\)

Answer:

Top row: \(x^2\), \(3x\), \(3\)

Middle row: \(-3x^3\), \(-9x^2\), \(-9x\)

Bottom row: \(-5x^2\), \(-15x\), \(-15\)

(The quotient is \(x^2 + 3x + 3\))