Factoring Trinomials Practice

Find constants whose sum is \(11\) and product is \(24\).

Expanding \((x+3)(x+8)\) gives leading coefficient \(1\), middle coefficient \(11\), and constant \(24\), exactly reproducing \(x^2+11x+24\).

Find constants whose sum is \(13\) and product is \(36\).

Expanding \((x+4)(x+9)\) gives leading coefficient \(1\), middle coefficient \(13\), and constant \(36\), exactly reproducing \(x^2+13x+36\).

Find constants whose sum is \(13\) and product is \(22\).

Expanding \((x+2)(x+11)\) gives leading coefficient \(1\), middle coefficient \(13\), and constant \(22\), exactly reproducing \(x^2+13x+22\).

Find constants whose sum is \(12\) and product is \(35\).

Expanding \((x+5)(x+7)\) gives leading coefficient \(1\), middle coefficient \(12\), and constant \(35\), exactly reproducing \(x^2+12x+35\).

Find constants whose sum is \(16\) and product is \(60\).

Expanding \((x+6)(x+10)\) gives leading coefficient \(1\), middle coefficient \(16\), and constant \(60\), exactly reproducing \(x^2+16x+60\).

Find constants whose sum is \(-10\) and product is \(21\).

Expanding \((x-3)(x-7)\) gives leading coefficient \(1\), middle coefficient \(-10\), and constant \(21\), exactly reproducing \(x^2-10x+21\).

Find constants whose sum is \(-10\) and product is \(24\).

Expanding \((x-4)(x-6)\) gives leading coefficient \(1\), middle coefficient \(-10\), and constant \(24\), exactly reproducing \(x^2-10x+24\).

Find constants whose sum is \(-11\) and product is \(18\).

Expanding \((x-2)(x-9)\) gives leading coefficient \(1\), middle coefficient \(-11\), and constant \(18\), exactly reproducing \(x^2-11x+18\).

Find constants whose sum is \(-13\) and product is \(40\).

Expanding \((x-5)(x-8)\) gives leading coefficient \(1\), middle coefficient \(-13\), and constant \(40\), exactly reproducing \(x^2-13x+40\).

Find constants whose sum is \(-17\) and product is \(70\).

Expanding \((x-7)(x-10)\) gives leading coefficient \(1\), middle coefficient \(-17\), and constant \(70\), exactly reproducing \(x^2-17x+70\).

Find constants whose sum is \(5\) and product is \(-24\).

Expanding \((x+8)(x-3)\) gives leading coefficient \(1\), middle coefficient \(5\), and constant \(-24\), exactly reproducing \(x^2+5x-24\).

Find constants whose sum is \(-5\) and product is \(-36\).

Expanding \((x-9)(x+4)\) gives leading coefficient \(1\), middle coefficient \(-5\), and constant \(-36\), exactly reproducing \(x^2-5x-36\).

Find constants whose sum is \(9\) and product is \(-22\).

Expanding \((x+11)(x-2)\) gives leading coefficient \(1\), middle coefficient \(9\), and constant \(-22\), exactly reproducing \(x^2+9x-22\).

Find constants whose sum is \(-2\) and product is \(-35\).

Expanding \((x-7)(x+5)\) gives leading coefficient \(1\), middle coefficient \(-2\), and constant \(-35\), exactly reproducing \(x^2-2x-35\).

Find constants whose sum is \(7\) and product is \(-60\).

Expanding \((x+12)(x-5)\) gives leading coefficient \(1\), middle coefficient \(7\), and constant \(-60\), exactly reproducing \(x^2+7x-60\).

Write the sum and product targets before listing candidate pairs.

The pair must add to \(17\) and multiply to \(ac=6(5)=30\). The values \(15\) and \(2\) satisfy both conditions.

Write the sum and product targets before listing candidate pairs.

The pair must add to \(-2\) and multiply to \(ac=8(-3)=-24\). The values \(6\) and \(-8\) satisfy both conditions.

Write the sum and product targets before listing candidate pairs.

The pair must add to \(13\) and multiply to \(ac=10(3)=30\). The values \(10\) and \(3\) satisfy both conditions.

Write the sum and product targets before listing candidate pairs.

The pair must add to \(-7\) and multiply to \(ac=12(-10)=-120\). The values \(8\) and \(-15\) satisfy both conditions.

Write the sum and product targets before listing candidate pairs.

The pair must add to \(17\) and multiply to \(ac=15(4)=60\). The values \(12\) and \(5\) satisfy both conditions.

Use \(ac=72\), split the middle term, and factor by grouping.

Expanding \((2x+3)(3x+4)\) gives leading coefficient \(6\), middle coefficient \(17\), and constant \(12\), exactly reproducing \(6x^2+17x+12\).

Use \(ac=-120\), split the middle term, and factor by grouping.

Expanding \((4x-5)(2x+3)\) gives leading coefficient \(8\), middle coefficient \(2\), and constant \(-15\), exactly reproducing \(8x^2+2x-15\).

Use \(ac=-210\), split the middle term, and factor by grouping.

Expanding \((5x+2)(3x-7)\) gives leading coefficient \(15\), middle coefficient \(-29\), and constant \(-14\), exactly reproducing \(15x^2-29x-14\).

Use \(ac=-60\), split the middle term, and factor by grouping.

Expanding \((6x+1)(2x-5)\) gives leading coefficient \(12\), middle coefficient \(-28\), and constant \(-5\), exactly reproducing \(12x^2-28x-5\).

Use \(ac=120\), split the middle term, and factor by grouping.

Expanding \((3x-4)(5x-2)\) gives leading coefficient \(15\), middle coefficient \(-26\), and constant \(8\), exactly reproducing \(15x^2-26x+8\).

A valid split preserves the original middle coefficient and has product equal to the leading coefficient times the constant.

The replacement coefficients \(15\) and \(4\) add to \(19\) and multiply to \(ac=60\).

A valid split preserves the original middle coefficient and has product equal to the leading coefficient times the constant.

The replacement coefficients \(8\) and \(-9\) add to \(-1\) and multiply to \(ac=-72\).

A valid split preserves the original middle coefficient and has product equal to the leading coefficient times the constant.

The replacement coefficients \(-35\) and \(2\) add to \(-33\) and multiply to \(ac=-70\).

A valid split preserves the original middle coefficient and has product equal to the leading coefficient times the constant.

The replacement coefficients \(-2\) and \(-35\) add to \(-37\) and multiply to \(ac=70\).

A valid split preserves the original middle coefficient and has product equal to the leading coefficient times the constant.

The replacement coefficients \(-12\) and \(5\) add to \(-7\) and multiply to \(ac=-60\).

After removing the GCF, reinspect the remaining trinomial instead of stopping early.

Extracting \(3\) leaves \(x^2+7x+10\), which factors as \((x+2)(x+5)\). Thus the complete factorization is \(3(x+2)(x+5)\).

After removing the GCF, reinspect the remaining trinomial instead of stopping early.

Extracting \(4\) leaves \(2x^2+5x-12\), which factors as \((2x-3)(x+4)\). Thus the complete factorization is \(4(2x-3)(x+4)\).

After removing the GCF, reinspect the remaining trinomial instead of stopping early.

Extracting \(5\) leaves \(3x^2-5x-2\), which factors as \((3x+1)(x-2)\). Thus the complete factorization is \(5(3x+1)(x-2)\).

After removing the GCF, reinspect the remaining trinomial instead of stopping early.

Extracting \(6\) leaves \(2x^2-x-15\), which factors as \((2x+5)(x-3)\). Thus the complete factorization is \(6(2x+5)(x-3)\).

After removing the GCF, reinspect the remaining trinomial instead of stopping early.

Extracting \(7\) leaves \(x^2-10x+24\), which factors as \((x-4)(x-6)\). Thus the complete factorization is \(7(x-4)(x-6)\).

A negative GCF can make the remaining leading coefficient positive, but every sign must be checked by expansion.

Extract \(-1\), then factor the remaining trinomial: \(-2x^2+5x+12=-(2x+3)(x-4)\). Expanding preserves the original leading sign.

A negative GCF can make the remaining leading coefficient positive, but every sign must be checked by expansion.

Extract \(-2\), then factor the remaining trinomial: \(-6x^2-26x+20=-2(3x-2)(x+5)\). Expanding preserves the original leading sign.

A negative GCF can make the remaining leading coefficient positive, but every sign must be checked by expansion.

Extract \(-3\), then factor the remaining trinomial: \(-3x^2+24x-36=-3(x-6)(x-2)\). Expanding preserves the original leading sign.

A negative GCF can make the remaining leading coefficient positive, but every sign must be checked by expansion.

Extract \(-1\), then factor the remaining trinomial: \(-8x^2+10x+3=-(4x+1)(2x-3)\). Expanding preserves the original leading sign.

A negative GCF can make the remaining leading coefficient positive, but every sign must be checked by expansion.

Extract \(-4\), then factor the remaining trinomial: \(-8x^2+12x+20=-4(2x-5)(x+1)\). Expanding preserves the original leading sign.

Check both the sum and product; do not force a factorization from one condition.

An integer factorization would require integers with sum \(2\) and product \(7\). Testing the factor pairs of \(7\) shows that none meet both conditions.

Check both the sum and product; do not force a factorization from one condition.

An integer factorization would require integers with sum \(4\) and product \(10\). Testing the factor pairs of \(10\) shows that none meet both conditions.

Check both the sum and product; do not force a factorization from one condition.

An integer factorization would require integers with sum \(-3\) and product \(8\). Testing the factor pairs of \(8\) shows that none meet both conditions.

Check both the sum and product; do not force a factorization from one condition.

An integer factorization would require integers with sum \(6\) and product \(11\). Testing the factor pairs of \(11\) shows that none meet both conditions.

Check both the sum and product; do not force a factorization from one condition.

An integer factorization would require integers with sum \(-5\) and product \(13\). Testing the factor pairs of \(13\) shows that none meet both conditions.

Use invariant checks instead of trusting the appearance of two binomials.

The AC target was computed incorrectly. A reliable correction checks the GCF, the \(ac\) product, the middle-coefficient sum, and the final expansion in that order.

Use invariant checks instead of trusting the appearance of two binomials.

Both the product and sum conditions must hold. A reliable correction checks the GCF, the \(ac\) product, the middle-coefficient sum, and the final expansion in that order.

Use invariant checks instead of trusting the appearance of two binomials.

The split or a grouping sign is incorrect. A reliable correction checks the GCF, the \(ac\) product, the middle-coefficient sum, and the final expansion in that order.

Use invariant checks instead of trusting the appearance of two binomials.

The expression is not completely factored. A reliable correction checks the GCF, the \(ac\) product, the middle-coefficient sum, and the final expansion in that order.

Use invariant checks instead of trusting the appearance of two binomials.

FOIL verification exposes an incorrect cross-term sum. A reliable correction checks the GCF, the \(ac\) product, the middle-coefficient sum, and the final expansion in that order.