Question

select the correct location in the expression.
consider this expression.
\\(\frac{4x^4 - 6x^3 + 6x + 3}{x - 3}\\)
which term in the quotient of this expression contains an error?
\\(4x^3 + 6x^2 + 18x + 78 + \frac{183}{x - 3}\\)

Explanation:

Step1: Perform Polynomial Long Division

Divide \(4x^4 - 4x^3 + 6x + 3\) by \(x - 3\).
- Divide \(4x^4\) by \(x\) to get \(4x^3\). Multiply \(x - 3\) by \(4x^3\): \(4x^4 - 12x^3\). Subtract from the dividend: \((4x^4 - 4x^3) - (4x^4 - 12x^3) = 8x^3\).
- Bring down \(6x + 3\) (wait, no, next term: divide \(8x^3\) by \(x\) to get \(8x^2\)? Wait, original dividend is \(4x^4 - 4x^3 + 0x^2 + 6x + 3\) (missing \(x^2\) term). Let's correct:
Dividend: \(4x^4 - 4x^3 + 0x^2 + 6x + 3\), Divisor: \(x - 3\).
1. \(4x^4 \div x = 4x^3\). Multiply divisor: \(4x^3(x - 3) = 4x^4 - 12x^3\). Subtract: \((4x^4 - 4x^3) - (4x^4 - 12x^3) = 8x^3\).
2. \(8x^3 \div x = 8x^2\). Multiply divisor: \(8x^2(x - 3) = 8x^3 - 24x^2\). Subtract: \((8x^3 + 0x^2) - (8x^3 - 24x^2) = 24x^2\).
3. \(24x^2 \div x = 24x\). Multiply divisor: \(24x(x - 3) = 24x^2 - 72x\). Subtract: \((24x^2 + 6x) - (24x^2 - 72x) = 78x\).
4. \(78x \div x = 78\). Multiply divisor: \(78(x - 3) = 78x - 234\). Subtract: \((78x + 3) - (78x - 234) = 237\).

Wait, the given quotient has \(6x^2\) (wrong, should be \(8x^2\) initially? No, wait original dividend's \(x^3\) term: \( -4x^3 - (-12x^3) = 8x^3\), so next term is \(8x^2\), not \(6x^2\). Wait, the given quotient is \(4x^3 + 6x^2 + 18x + 78\). Let's check the \(x^2\) term:

In the given quotient, after \(4x^3\), the next term is \(6x^2\). But from division, after \(4x^3\), we get \(8x^3\) (after subtraction), so \(8x^3 \div x = 8x^2\), not \(6x^2\). Wait, maybe I made a mistake. Wait original dividend: \(4x^4 - 4x^3 + 6x + 3\) (no \(x^2\) term). Let's redo with correct steps:

1. \(4x^4 \div x = 4x^3\). Multiply \(x - 3\) by \(4x^3\): \(4x^4 - 12x^3\). Subtract from dividend: \((4x^4 - 4x^3) - (4x^4 - 12x^3) = 8x^3\).
2. Now, divide \(8x^3\) by \(x\) to get \(8x^2\). Multiply \(x - 3\) by \(8x^2\): \(8x^3 - 24x^2\). Subtract: \((8x^3 + 0x^2) - (8x^3 - 24x^2) = 24x^2\).
3. Divide \(24x^2\) by \(x\) to get \(24x\). Multiply \(x - 3\) by \(24x\): \(24x^2 - 72x\). Subtract: \((24x^2 + 6x) - (24x^2 - 72x) = 78x\).
4. Divide \(78x\) by \(x\) to get \(78\). Multiply \(x - 3\) by \(78\): \(78x - 234\). Subtract: \((78x + 3) - (78x - 234) = 237\).

So the correct quotient terms (before remainder) should be \(4x^3 + 8x^2 + 24x + 78\), with remainder \(237\).

The given quotient has \(6x^2\) (instead of \(8x^2\)) and \(18x\) (instead of \(24x\)), and remainder \(183\) (instead of \(237\)). But the question is which term in the quotient (the polynomial part, not remainder) has an error. The \(6x^2\) term is wrong (should be \(8x^2\)), \(18x\) is wrong (should be \(24x\)), but let's check the given quotient: \(4x^3 + 6x^2 + 18x + 78\).

Wait, when we do the division, after \(4x^3\), the next term ( \(x^2\) term) should be \(8x^2\) (from \(8x^3 \div x\)), but given is \(6x^2\). Alternatively, maybe the error is in the \(18x\) term? Wait, let's check the multiplication for the \(x\) term:

If we use the given \(6x^2\) term: Multiply \(x - 3\) by \(6x^2\): \(6x^3 - 18x^2\). Subtract from \(8x^3\) (from step 1): \(8x^3 - (6x^3 - 18x^2) = 2x^3 + 18x^2\). Then divide \(2x^3\) by \(x\) to get \(2x^2\)? No, this is confusing. The key is that the correct \(x^2\) term after \(4x^3\) should be \(8x^2\) (from \(8x^3 \div x\)), but the given is \(6x^2\). Alternatively, the \(18x\) term: let's check the linear term.

Wait, the given quotient has \(18x\). From our division, after \(24x^2\) (if we did step 2 correctly), we get \(78x\) (step 3). Wait, maybe the original problem's dividend was \(4x^4 - 4x^3 + 6x^2 + 6x + 3\)? No, the problem states \(4x^4 - 4x^3 + 6x + 3\) (missing \(x^2\) term).

But the main error in the quotient terms (the polynomial part: \(4x^3 + 6x^2 + 18x + 78\)) is the \(6x^2\) term (should be \(8x^2\)) or \(18x\) (should be \(24x\)), or \(78\) (but \(78\) is correct in our division? Wait no, in our division, the constant term was \(78\), but the remainder was \(237\). Wait, no:

Wait, dividend: \(4x^4 - 4x^3 + 0x^2 + 6x + 3\)
Divisor: \(x - 3\)

1. \(4x^4 \div x = 4x^3\). Multiply divisor: \(4x^3(x - 3) = 4x^4 - 12x^3\). Subtract: \( (4x^4 - 4x^3) - (4x^4 - 12x^3) = 8x^3\).
2. \(8x^3 \div x = 8x^2\). Multiply divisor: \(8x^2(x - 3) = 8x^3 - 24x^2\). Subtract: \( (8x^3 + 0x^2) - (8x^3 - 24x^2) = 24x^2\).
3. \(24x^2 \div x = 24x\). Multiply divisor: \(24x(x - 3) = 24x^2 - 72x\). Subtract: \( (24x^2 + 6x) - (24x^2 - 72x) = 78x\).
4. \(78x \div x = 78\). Multiply divisor: \(78(x - 3) = 78x - 234\). Subtract: \( (78x + 3) - (78x - 234) = 237\).

So the correct quotient (polynomial part) is \(4x^3 + 8x^2 + 24x + 78\), with remainder \(237\). The given quotient has \(6x^2\) (instead of \(8x^2\)) and \(18x\) (instead of \(24x\)), and remainder \(183\) (instead of \(237\)). But the question is which term in the quotient (the polynomial terms: \(4x^3, 6x^2, 18x, 78\)) has an error. The \(6x^2\) term is incorrect (should be \(8x^2\)), \(18x\) is incorrect (should be \(24x\)), but let's check the multiplication for the \(x^2\) term:

If we use the given \(6x^2\) term:
Multiply \(x - 3\) by \(6x^2\): \(6x^3 - 18x^2\).
Subtract from \(8x^3\) (from step 1): \(8x^3 - (6x^3 - 18x^2) = 2x^3 + 18x^2\).
Then divide \(2x^3\) by \(x\) to get \(2x^2\) (not \(18x\)). This is inconsistent.

Alternatively, the error is in the \(18x\) term. Wait, the given quotient's \(18x\) term: if we multiply \(x - 3\) by \(18x\), we get \(18x^2 - 54x\). Subtract from \(24x^2 + 6x\) (from step 2, if we had \(24x^2\)): \(24x^2 + 6x - (18x^2 - 54x) = 6x^2 + 60x\). Then divide \(6x^2\) by \(x\) to get \(6x\), which is not \(78\).

The most probable error is the \(6x^2\) term (should be \(8x^2\)) or \(18x\) (should be \(24x\)). But let's check the original problem again. The given quotient is \(4x^3 + 6x^2 + 18x + 78\). From our division, the correct \(x^2\) term is \(8x^2\), so \(6x^2\) is wrong. The correct linear term is \(24x\), so \(18x\) is wrong. However, the question asks "which term in the quotient" (the polynomial part, not remainder). The \(6x^2\) term is incorrect (first error in the quotient terms after \(4x^3\)).

Step2: Identify the Erroneous Term

The quotient terms are \(4x^3\) (correct, as \(4x^4 \div x = 4x^3\)), \(6x^2\) (incorrect, should be \(8x^2\)), \(18x\) (incorrect, should be \(24x\)), \(78\) (correct in our division, but remainder is wrong). But the key is that the \(6x^2\) term is the first error in the quotient's polynomial terms. Alternatively, the \(18x\) term. Wait, let's check with synthetic division.

Synthetic division with root \(x = 3\):

Coefficients: \(4, -4, 0, 6, 3\) (for \(4x^4 - 4x^3 + 0x^2 + 6x + 3\)).

Bring down \(4\).
Multiply by \(3\): \(12\). Add to \(-4\): \(8\).
Multiply by \(3\): \(24\). Add to \(0\): \(24\).
Multiply by \(3\): \(72\). Add to \(6\): \(78\).
Multiply by \(3\): \(234\). Add to \(3\): \(237\).

So the quotient polynomial is \(4x^3 + 8x^2 + 24x + 78\), remainder \(237\).

Comparing to given quotient: \(4x^3 + 6x^2 + 18x + 78\).

- \(4x^3\): correct.
- \(6x^2\): should be \(8x^2\) (error).
- \(18x\): should be \(24x\) (error).
- \(78\): correct (constant term of quotient).

The question asks "which term in the quotient" (the polynomial part). The \(6x^2\) term is incorrect (first error in the \(x^2\) term), or \(18x\) (linear term). But the most probable is the \(6x^2\) term (since the \(x^2\) coefficient from synthetic division is \(8\), not \(6\)).

Answer:

The term \(6x^2\) (or \(18x\)) in the quotient contains an error. Specifically, the \(6x^2\) term should be \(8x^2\), and the \(18x\) term should be \(24x\). However, the most prominent error in the quotient's polynomial terms is \(6x^2\) (or \(18x\); depending on the problem's intent, but from synthetic division, the \(x^2\) term is \(8x^2\), so \(6x^2\) is wrong).