2.1.4
Intuition Hum Expressions Ko Multiply Kyun Karte HainAlgebraic expressions ki multiplication basically variables ke saath repeated addition hai. Jab tum 3 ( x + 2 ) ( x + 2 )
Definition Distributive PropertyKisi bhi number ya expression a b c a ( b + c ) = ab + a c kisi single term ko ek sum se multiply karne ke liye, us term ko har addend se alag-alag multiply karo, phir results ko add karo .
Yeh kyun kaam karta hai: Socho a ( b + c ) a b a c
Yeh step-by-step kyun? Polynomial ka har term independent hota hai. Monomial ko har ek term ko alag-alag "touch" karna padta hai kyunki addition, multiplication mein shortcuts allow nahi karta.
Worked example Example 1: Monomial × BinomialMultiply karo: 3 x ( 2 x + 5 )
Step 1: Monomial identify karo: 3 x Step 2: Distributive property apply karo
3 x ⋅ 2 x + 3 x ⋅ 5 Yeh step kyun? Humein 3 x
Step 3: Coefficients multiply karo aur like bases ke liye exponents add karo
= 6 x 2 + 15 x
x 2 x 1 ⋅ x 1 = x 1 + 1 = x 2
✓ Answer: 6 x 2 + 15 x
Worked example Example 2: Monomial × TrinomialMultiply karo: − 2 a 2 ( 3 a 2 − 4 a + 7 )
Step 1: − 2 a 2 = ( − 2 a 2 ) ( 3 a 2 ) + ( − 2 a 2 ) ( − 4 a ) + ( − 2 a 2 ) ( 7 )
Har term alag-alag kyun? Distributive property humse require karti hai ki monomial ko har addend se multiply karein.
Step 2: Har product multiply karo
Pehla term: ( − 2 ) ( 3 ) ⋅ a 2 + 2 = − 6 a 4
Doosra term: ( − 2 ) ( − 4 ) ⋅ a 2 + 1 = 8 a 3
Teesra term: ( − 2 ) ( 7 ) ⋅ a 2 = − 14 a 2
Yeh signs kyun? Negative × positive = negative; negative × negative = positive
✓ Answer: − 6 a 4 + 8 a 3 − 14 a 2
General principle: Pehli polynomial ka har term, doosri polynomial ke har term ko multiply karta hai .
Worked example Example 3: Binomial × Binomial (FOIL)Multiply karo: ( 2 x + 3 ) ( x − 4 )
FOIL use karke:
F irst: 2 x ⋅ x = 2 x 2 O uter: 2 x ⋅ ( − 4 ) = − 8 x I nner: 3 ⋅ x = 3 x L ast: 3 ⋅ ( − 4 ) = − 12 FOIL kyun? Yeh ensure karta hai ki hum koi bhi term-by-term multiplication miss na karein.
Step 2: Like terms combine karo
2 x 2 + ( − 8 x + 3 x ) − 12 = 2 x 2 − 5 x − 12
Combine kyun? − 8 x 3 x
✓ Answer: 2 x 2 − 5 x − 12
Worked example Example 4: Binomial × TrinomialMultiply karo: ( x + 2 ) ( x 2 − 3 x + 5 )
Step 1: x x ( x 2 − 3 x + 5 ) = x 3 − 3 x 2 + 5 x
Pehle yeh kyun? Hum x
Step 2: 2 2 ( x 2 − 3 x + 5 ) = 2 x 2 − 6 x + 10
Step 3: Saare products add karo
x 3 − 3 x 2 + 5 x + 2 x 2 − 6 x + 10 Step 4: Like terms combine karo
= x 3 + ( − 3 x 2 + 2 x 2 ) + ( 5 x − 6 x ) + 10 = x 3 − x 2 − x + 10 Combine kyun? Simplification ke liye identical variable parts wale terms ko group karna zaroori hai.
✓ Answer: x 3 − x 2 − x + 10
Worked example Example 5: Trinomial × Binomial with NegativesMultiply karo: ( 2 a − 3 b + 1 ) ( a + 4 b )
Yeh form kyun? Yahan trinomial ( 2 a − 3 b + 1 ) 2 a − 3 b 1 ( a + 4 b )
Step 1: 2 a 2 a ( a + 4 b ) = 2 a 2 + 8 ab
Step 2: − 3 b − 3 b ( a + 4 b ) = − 3 ab − 12 b 2
Negative kyun? − 3 b ⋅ 4 b = − 12 b 2
Step 3: 1 1 ( a + 4 b ) = a + 4 b
Step 4: Saare products combine karo
2 a 2 + 8 ab − 3 ab − 12 b 2 + a + 4 b = 2 a 2 + 5 ab − 12 b 2 + a + 4 b
5 ab 8 ab − 3 ab = 5 ab
✓ Answer: 2 a 2 + 5 ab − 12 b 2 + a + 4 b
Mnemonic Binomials ke liye FOIL"First Outer Inner Last"
Ek book padhne ki tarah socho: First letters se shuru karo, phir Outer edges par jao, phir Inner close karo, phir Last ends.
General polynomial multiplication ke liye: "Every term with Every term" — ek handshake tournament ki tarah socho jahan ek group ka har banda doosre group ke har bande se haath milata hai.
Common mistake Mistake 1: Sabhi Terms Mein Distribute Karna Bhool JanaGalat: 3 ( x + y + z ) = 3 x + y + z Yeh sahi kyun lagta hai: Students aksar pehle term mein distribute karte hain aur baaki bhool jaate hain, jaisa ki parentheses "kamzor" barriers hain.
Steel-man argument: "Maine saamne wale number ko multiply kar diya, toh ho gaya."
Fix: Distributive property require karti hai ki andar ke HAR term se multiply karo . Aise socho: parentheses ek "package deal" banate hain — coefficient ko andar ki har cheez par apply karna hota hai.
Sahi: 3 ( x + y + z ) = 3 x + 3 y + 3 z
Common mistake Mistake 2: Galat Exponent AdditionGalat: ( x 2 ) ( x 3 ) = x 5 ( x 2 ) 3 = x 5 Yeh sahi kyun lagta hai: Students powers ki multiplication aur power of a power ko confuse karte hain.
Fix:
Multiplication: x a ⋅ x b = x a + b
Power of power: ( x a ) b = x a ⋅ b
Sahi: ( x 2 ) 3 = x 2 ⋅ 3 = x 6
Common mistake Mistake 3: Negatives ke Saath Sign ErrorsGalat: ( x − 3 ) ( x − 2 ) = x 2 − 6
Yeh sahi kyun lagta hai: Students constants multiply karte hain (3 × 2 = 6
Fix: FOIL dhyan se use karo:
F: x ⋅ x = x 2
O: x ⋅ ( − 2 ) = − 2 x
I: ( − 3 ) ⋅ x = − 3 x
L: ( − 3 ) ⋅ ( − 2 ) = + 6
Sahi: ( x − 3 ) ( x − 2 ) = x 2 − 2 x − 3 x + 6 = x 2 − 5 x + 6
Common mistake Mistake 4: "Squaring matlab Doubling"Galat: ( x + 3 ) 2 = x 2 + 9
Yeh sahi kyun lagta hai: Students har term ko alag-alag square kar dete hain, middle term bhool jaate hain.
Steel-man: "Agar main dono parts ko square karun, toh poori cheez ka square milna chahiye."
Fix: ( x + 3 ) 2 = ( x + 3 ) ( x + 3 ) a 2 + 2 ab + b 2
Sahi: ( x + 3 ) 2 = x 2 + 6 x + 9
Recall Ek 12-Saal ke Bache Ko Explain KaroSocho tumhare paas candies ke bags hain. Agar ek bag mein ( x + 5 ) 3 3 ( x + 5 ) 3 x 3 x 3 5 3 ( x + 5 ) = 3 x + 15 ( x + 2 ) ( x + 3 ) x × x
x x x ⋅ x = x 2 x 3 x ⋅ 3 = 3 x 2 x 2 ⋅ x = 2 x 2 3 2 ⋅ 3 = 6 x 2 + 3 x + 2 x + 6 = x 2 + 5 x + 6
Distributive Property — saari multiplication expansion ki foundationCombining Like Terms — polynomials multiply karne ke baad zaroori haiFactoring Polynomials — polynomial multiplication ka reverse processExponent Rules — same base wale terms multiply karte waqt use hote hainQuadratic Expressions — aksar binomial multiplication se result hote hainPascal's Triangle — ( a + b ) n Area Models for Algebra — polynomial multiplication ki visual representation
Structure identify karo: monomial × polynomial hai ya polynomial × polynomial?Systematically distribute karo: ensure karo ki pehle expression ka har term doosre ke har term ko multiply kareExponent rules apply karo: jab bases same hon toh exponents add karoSigns dhyan se track karo: negative × negative = positive, negative × positive = negativeLike terms combine karo: identical variable parts wale terms ko add/subtract karke simplify karoVerify karo: ek simple number (jaise x = 1
#flashcards/maths
Distributive property kya hai? a ( b + c ) = ab + a c
Monomial ko polynomial se kaise multiply karte hain? Monomial ko polynomial ke har term se alag-alag multiply karo, phir saare products add karo
Binomial multiplication mein FOIL ka full form kya hai? First, Outer, Inner, Last — ( a + b ) ( c + d )
Polynomial × polynomial multiplication ka general rule kya hai? Pehli polynomial ka har term, doosri polynomial ke har term ko multiply karna chahiye
( a + b ) 2 ( a + b ) ( a + b ) = a 2 + ab + ba + b 2 = a 2 + 2 ab + b 2
Difference of squares formula kya hai? ( a + b ) ( a − b ) = a 2 − b 2
( x + 3 ) 2 = x 2 + 9 Kyunki ( x + 3 ) 2 = ( x + 3 ) ( x + 3 ) = x 2 + 6 x + 9 2 ab
x 2 ⋅ x 3 Unhe add karo: x 2 ⋅ x 3 = x 2 + 3 = x 5
2 x ( 3 x − 5 ) 2 x ⋅ 3 x + 2 x ⋅ ( − 5 ) = 6 x 2 − 10 x
( x + 4 ) ( x − 2 ) x 2 − 2 x + 4 x − 8 = x 2 + 2 x − 8
( x − 3 ) 2 Middle term bhool jana aur x 2 + 9 x 2 − 6 x + 9
Combine karne se pehle binomial ko trinomial se multiply karne par kitne terms milte hain? 2 × 3 = 6
Distributive Property a b+c = ab+ac
Geometric intuition: a groups
Exponent rule: add exponents
Multiply each term separately
Treat first poly as chunk
