Ko'phadni ko'paytirish

Bu nima va nega kerak

Fermaning yer maydoni uzunligi $(a + 5)$ metr, eni $(a - 2)$ metr bo'lsa, yuzi qancha? Bu masalada $S = (a+5)(a-2)$ — ikki ko'phadning ko'paytmasi.

Ko'phadlarni ko'paytirish — qisqa ko'paytirish formulalarining, kvadrat tenglamalarning va ko'plab murakkab masalalarning asosi. Bu amalni tez va to'g'ri bajara bilmasdan, keyingi darslarda qiynalish muqarrar.

Asosiy g'oya

Birhadni ko'phadga ko'paytirish — taqsimot qonuni:

$a(b + c) = ab + ac.$

Ya'ni: birhad ko'phadning har bir hadiga ko'paytiriladi.

Ko'phadni ko'phadga ko'paytirish — har bir had har bir hadga:

$(a + b)(c + d) = ac + ad + bc + bd.$

FOIL usuli (inglizcha: First, Outer, Inner, Last) — ikki hadli ko'phadlar uchun qulay:

• First (birinchi × birinchi): $a \cdot c$
• Outer (tashqi × tashqi): $a \cdot d$
• Inner (ichki × ichki): $b \cdot c$
• Last (oxirgi × oxirgi): $b \cdot d$

Ko'paytirish qadamlari:

1. Birinchi ko'phadning har bir hadini ikkinchi ko'phadning har bir hadiga ko'paytir.
2. Hosil bo'lgan ifodani yoz.
3. O'xshash hadlarni yig'ib, standart ko'rinishga keltir.

Yechilgan misollar

1-misol. Birhadni ko'phadga ko'paytirish

$4x(3x^2 - 2x + 1)$ ni oching.

• $4x \cdot 3x^2 = 12x^3$
• $4x \cdot (-2x) = -8x^2$
• $4x \cdot 1 = 4x$

Javob: $12x^3 - 8x^2 + 4x$.

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2-misol. Ikki hadli ko'phadlarni ko'paytirish

$(x + 5)(x - 3)$ ni oching.

FOIL usuli:

• F: $x \cdot x = x^2$
• O: $x \cdot (-3) = -3x$
• I: $5 \cdot x = 5x$
• L: $5 \cdot (-3) = -15$

Yig'amiz: $x^2 - 3x + 5x - 15 = x^2 + 2x - 15$.

Tekshirish ($x = 1$): $(1+5)(1-3) = 6 \cdot (-2) = -12$. Formula: $1 + 2 - 15 = -12$ ✓.

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3-misol. Uch hadli ko'phadga ko'paytirish

$(2a - 1)(a^2 + 3a - 2)$ ni oching.

• $2a \cdot a^2 = 2a^3$
• $2a \cdot 3a = 6a^2$
• $2a \cdot (-2) = -4a$
• $(-1) \cdot a^2 = -a^2$
• $(-1) \cdot 3a = -3a$
• $(-1) \cdot (-2) = 2$

Yig'amiz: $2a^3 + (6a^2 - a^2) + (-4a - 3a) + 2 = 2a^3 + 5a^2 - 7a + 2$.

Javob: $2a^3 + 5a^2 - 7a + 2$.

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4-misol. Manfiy birhad bilan ko'paytirish

$-3b^2(2b^2 - b + 4)$ ni oching.

• $-3b^2 \cdot 2b^2 = -6b^4$
• $-3b^2 \cdot (-b) = 3b^3$
• $-3b^2 \cdot 4 = -12b^2$

Javob: $-6b^4 + 3b^3 - 12b^2$.

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5-misol. Murakkab misol: uch ko'phadni ko'paytirish

$x(x+1)(x-1)$ ni oching.

• Avval ichkisini: $(x+1)(x-1) = x^2 - x + x - 1 = x^2 - 1$.
• Keyin: $x(x^2 - 1) = x^3 - x$.

Javob: $x^3 - x$.

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Mashq

1. $5a(a - 3)$ ni oching.
2. $-2x(x^2 + 4x - 1)$ ni oching.
3. $(m + 4)(m - 4)$ ni hisoblang.
4. $(2x + 1)(3x - 2)$ ni oching.
5. $(a - b)(a + 2b)$ ni oching.
6. $(x + 3)^2$ ni — ya'ni $(x+3)(x+3)$ ni — oching.
7. $(2n - 3)(n^2 + n - 2)$ ni hisoblang.
8. $3t(t-1)(t+1)$ ni oching.
9. $(p + q)(p - q)(p^2 + q^2)$ ni soddalashtiring.
10. Aylananing radiusi $(r + 1)$. Yuzasini $\pi(r+1)^2$ formulasi orqali $\pi$ ni tashqarida qoldirib oching.

Javoblar va yechim yo'llari

1. $5a^2 - 15a$.
2. $-2x^3 - 8x^2 + 2x$.
3. $m^2 - 16$. (Kvadratlar ayirmasi formulasi.)
4. $6x^2 - 4x + 3x - 2 = 6x^2 - x - 2$.
5. $a^2 + 2ab - ab - 2b^2 = a^2 + ab - 2b^2$.
6. $x^2 + 3x + 3x + 9 = x^2 + 6x + 9$.
7. $2n^3 + 2n^2 - 4n - 3n^2 - 3n + 6 = 2n^3 - n^2 - 7n + 6$.
8. $(t-1)(t+1) = t^2 - 1$; $3t(t^2-1) = 3t^3 - 3t$.
9. $(p+q)(p-q) = p^2-q^2$; $(p^2-q^2)(p^2+q^2) = p^4-q^4$.
10. $\pi(r^2+2r+1)$ — ya'ni $\pi r^2+2\pi r+\pi$.

Keng tarqalgan xatolar

1. Birhadni faqat birinchi hadga ko'paytirish. ✗ $3x(2x + 1) = 6x^2 + 1$. ✓ $3x(2x + 1) = 6x^2 + 3x$. (Ikkinchi hadni ham ko'paytirish kerak.)

2. Ko'paytmada o'xshash hadlarni qo'shmaslik. ✗ $(x+2)(x+3) = x^2 + 2x + 3x + 6 = x^2 + 2x + 3x + 6$. ✓ $x^2 + 5x + 6$. (O'xshash hadlarni yig'amiz.)

3. Manfiy birhad bilan ko'paytirganda ishorani unutish. ✗ $-2a(a - 3) = -2a^2 - 6a$. ✓ $-2a \cdot a = -2a^2$; $-2a \cdot (-3) = +6a$. Javob: $-2a^2 + 6a$.

4. $(x+a)^2 = x^2 + a^2$ deb yozish. ✗ $(x + 3)^2 = x^2 + 9$. ✓ $(x+3)^2 = x^2 + 6x + 9$. (O'rta had — $2 \cdot x \cdot 3 = 6x$ — unutildi.)

5. Had sonini noto'g'ri hisoblash. ✗ $(a + b)(c + d) = 3$ had hosil qiladi. ✓ Ikki hadli × ikki hadli = 4 had, keyin o'xshashlarni yig'amiz.

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Keyingi qadam

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