Class 8 · English Version · Mathematics
2026 Class 8 Mathematics · English Version · Chapter 4 · Chapter companion part
Choose a factor method, split signed middle terms, handle substituted quadratics and verify the result by expansion.
Class 8 English Version: part C of the chapter 4 companion. Eight sections and 30 written tasks. Selected companion coverage, not complete textbook coverage; use all four parts and compare the mapped source methods. The software checks stated numerical values or coefficients; symbolic work, equivalent forms and proofs need guardian or teacher review.
Open this topic in Skills Lab →
A factor divides an expression exactly in the chosen polynomial setting. Factoring reverses multiplication. First extract a common numerical/variable factor: 12x³y² + 18x²y = 6x²y(2xy + 3). Additive terms are not automatically factors; x + 2 is a sum, while x(x + 2) is a product. Expanding the proposed product checks the whole expression.
For ax + ay + bx + by, group a(x + y) + b(x + y), then factor (a + b)(x + y). Terms may be reordered, but signs must move with them. If two grouped brackets are opposites, factor out −1 from one bracket before combining. Do not discard an unmatched term to manufacture a common bracket.
9x² − 12xy + 4y² = (3x − 2y)². A complete-square expression can also be followed by a subtraction: x² + 4xy + 4y² − 9z² = (x + 2y − 3z)(x + 2y + 3z). Adding and subtracting the same expression does not change a polynomial; use it transparently to complete a square rather than insert an unmatched term.
2x⁴ + 54xy³ = 2x(x³ + 27y³) = 2x(x + 3y)(x² − 3xy + 9y²). For x⁶ − y⁶, start with a difference of squares or cubes and continue: (x − y)(x + y)(x² + xy + y²)(x² − xy + y²). Say over which coefficient domain the factorisation is complete; do not stop at an expression that still has a known factor pattern.
Monic means the coefficient of x² is 1. For x² + px + q, find signed integers m,n with m + n = p and mn = q; then factor (x + m)(x + n). Positive q gives equal signs, chosen by the sum; negative q gives opposite signs. For x² + 9x + 20, choose 4 and 5. For x² − 9x + 20, choose −4 and −5. Checking the product alone is insufficient.
For ax² + bx + c with a ≠ 0, seek m + n = b and mn = ac. In 3x² + 11x + 6, ac = 18 and 2 + 9 = 11. Rewrite 3x² + 2x + 9x + 6 = x(3x + 2) + 3(3x + 2) = (3x + 2)(x + 3). The split numbers are middle-term coefficients; they do not necessarily become the constants in two monic factors. With p as the variable and q as a parameter, 2p² − 7pq + 3q² splits as 2p² − 6pq − pq + 3q² and factors as (2p − q)(p − 3q).
For (x² − x)² + 2(x² − x) − 8, set u = x² − x. Then u² + 2u − 8 = (u + 4)(u − 2). Substitute back and continue: (x² − x + 4)(x − 2)(x + 1). Parameter expressions also serve as coefficients: x² + (3a + 4b)x + (2a² + 5ab + 3b²) factors as (x + a + b)(x + 2a + 3b). Verify both the linear sum and the constant product.
Integer-pair search does not always succeed: x² + 1 has no integer linear factors. The pairs with integer product 1 are (1,1) and (−1,−1); neither sum is 0. An identity such as ax² + (a² + 1)x + a = (ax + 1)(x + a) holds even at a = 0, but it is then not a quadratic in x. Expanding is a general verification; trying one numerical input can miss an incorrect factorisation.
For 4x² + 8x + 3, ac = 12 and split 8 as 6 + 2. Then 4x² + 6x + 2x + 3 = 2x(2x + 3) + (2x + 3) = (2x + 1)(2x + 3). Expansion recovers 4x² + 8x + 3.
A reviewer should accept reordered factors and paired sign changes when their product is unchanged. An exact string match cannot fairly grade all such answers.
If no integer pair works, recheck the coefficients and domain. Do not fabricate a pair or silently switch to complex factors.
Original examples of monic quadratic middle-term pairs.
| Polynomial | Pair | Sum / product |
|---|---|---|
| x² + 9x + 20 | 4, 5 | 9 / 20 |
| x² − 9x + 20 | −4, −5 | −9 / 20 |
| x² + 2x − 15 | −3, 5 | 2 / −15 |
| x² − 2x − 15 | −5, 3 | −2 / −15 |
For negative constant −15, the signs are opposite. Choose their magnitudes and signs so the sum also matches; multiplying to −15 alone is not enough.
Factoring reverses multiplication without dropping any term.
12x³y² + 18x²y = 6x²y(2xy + 3) ax + ay + bx + by = (a + b)(x + y) x² + 4xy + 4y² − 9z² = (x + 2y − 3z)(x + 2y + 3z)
A matching value at one chosen input does not prove equality of polynomials. Show distribution or a valid identity.
For ax² + bx + c, use a ≠ 0 and a signed pair whose product is ac.
mn = ac; m + n = b 3x² + 11x + 6 = 3x² + 2x + 9x + 6 = x(3x + 2) + 3(3x + 2) = (3x + 2)(x + 3)
The split pair need not equal the constants in two monic factors. If no integer pair works, recheck the input and coefficient domain.
A temporary variable represents the same expression everywhere.
u = x² − x u² + 2u − 8 = (u + 4)(u − 2) = (x² − x + 4)(x² − x − 2) = (x² − x + 4)(x − 2)(x + 1)
Do not leave u in the final answer unless the task asks for it. Completeness is relative to integer coefficients in this example.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original proportionate rectangle at the explicitly chosen x = 2. The drawing is not a physical model for negative side lengths.
Match each letter to its explanation. Read the quantities, units and relationships, then check the counts or equations. These are original teaching models.
Original symbolic guide. Arrows show a reasoning sequence, not mandatory trial steps for every expression.
Show your working on paper. Check your answer, then choose Next.
6
The greatest common divisor of coefficients 12 and 18 is 6.
2
The common exponent is min(3,2) = 2.
4
4 + 5 = 9 and 4 × 5 = 20.
5
The larger addend is 5.
-5
−5 + (−4) = −9 and their product is 20.
-3
−3 + 5 = 2 and (−3) × 5 = −15.
-5
−5 + 3 = −2 and (−5) × 3 = −15.
18
a = 3 and c = 6, so ac = 18.
2
The pair is 2 and 9.
2
Expanding (3x + 2)(x + 3) gives 3x² + 11x + 6.
-6
−6 and −1 sum to −7 and multiply to 6.
4
4 − 2 = 2 and 4 × (−2) = −8.
24
(5 − 1)(5 + 1) = 4 × 6 = 24.
63
2⁶ − 1 = 64 − 1 = 63.
0
The pairs (1,1) and (−1,−1) have sums 2 and −2; neither qualifies.
Write in your own words. A guardian or teacher should review this response.
6x²y(2xy + 3). The common coefficient is 6; minimum exponents are x² and y.
Expand the result and retain both original terms.
Write in your own words. A guardian or teacher should review this response.
a(x + y) + b(x + y) = (a + b)(x + y).
Reordering is allowed; every sign and term must be preserved.
Write in your own words. A guardian or teacher should review this response.
(3x − 2y)². The middle term is −2 × 3x × 2y.
Check both outer squares and the signed middle product.
Write in your own words. A guardian or teacher should review this response.
(x + 2y)² − (3z)² = (x + 2y − 3z)(x + 2y + 3z).
Show the complete-square stage before using conjugates.
Write in your own words. A guardian or teacher should review this response.
2x(x³ + 27y³) = 2x(x + 3y)(x² − 3xy + 9y²).
Do not leave the recognised cube sum unfactored.
Write in your own words. A guardian or teacher should review this response.
(x − y)(x + y)(x² + xy + y²)(x² − xy + y²). Start with (x³ − y³)(x³ + y³), then factor both cubes.
Expand paired factors or apply both cube identities to verify.
Write in your own words. A guardian or teacher should review this response.
The first is (x + 4)(x + 5); the second is (x − 4)(x − 5). Both constant products are 20, while the sums are 9 and −9.
Require sum and product checks for both rows.
Write in your own words. A guardian or teacher should review this response.
The first is (x − 3)(x + 5); the second is (x − 5)(x + 3). The products are negative and the larger-magnitude signed addend determines the sum.
Check both signed sums as well as products.
Write in your own words. A guardian or teacher should review this response.
3x² + 2x + 9x + 6 = x(3x + 2) + 3(3x + 2) = (3x + 2)(x + 3).
The split integers multiply to ac = 18, not merely c = 6.
Write in your own words. A guardian or teacher should review this response.
2p² − 6pq − pq + 3q² = 2p(p − 3q) − q(p − 3q) = (2p − q)(p − 3q).
Keep the parameter and the minus sign when extracting the second group.
Write in your own words. A guardian or teacher should review this response.
With u = x² − x, factor (u + 4)(u − 2). Substitute back: (x² − x + 4)(x² − x − 2) = (x² − x + 4)(x − 2)(x + 1).
The remaining quadratic has no integer pair with product 4 and sum −1.
Write in your own words. A guardian or teacher should review this response.
(x + a + b)(x + 2a + 3b). The constant product (a + b)(2a + 3b) is 2a² + 5ab + 3b².
Verify the full coefficient sum and constant product, treating a,b as parameters.
Write in your own words. A guardian or teacher should review this response.
(ax + 1)(x + a). Expansion proves it for every a. At a = 0 it becomes x, so the identity holds but the expression is no longer quadratic in x.
Do not divide by a without stating a ≠ 0.
Write in your own words. A guardian or teacher should review this response.
This is (a − b)³ − b³, so it equals (a − 2b)(a² − ab + b²).
Show the extra −b³ and expand to check the final three mixed coefficients.
Write in your own words. A guardian or teacher should review this response.
For a monic integer factorisation, the constants must multiply to ±1 and sum to 0. Product +1 allows only (1,1) or (−1,−1), neither with sum 0. Product −1 allows (1,−1), giving (x + 1)(x − 1).
State the integer-coefficient domain rather than saying no factorisation exists in any number system.
Create a factor-method decision card and five fully worked factorisations, including a substitution.
Keep your work on paper or in a local file. This page does not upload it or automatically grade the project.
This original Kidemy practice follows the topic order of 2026 Class 8 Mathematics · English Version, chapter 4, printed pages 48–76. It does not replace the complete chapter or its practical teaching.
Official book catalogue and source →
Teacher-led lessons, schedules and prices for this outline are upcoming. You can use any starter practice above now without booking.
Solo practice works without an account. Sign in to save a practice record to your learner profile.
Exact lesson links and related activities are labelled separately. Return-to-course links retain your selected learning path.
Play this exact lesson in Skills Lab →