TODAY I WILL SHOW YOU SOME AMAZING MATHEMATICAL QUESTIONS (IN THE FIELD OF FACTORIZATION) THAT WILL AMAZE YOU.
THESE QUESTIONS CAN BE UNDERSTOOD BY PEOPLE OF AGES 12 AND ABOVE. SO LET'S START.
Q1) FIND THE FACTORS OF ⟶ (a-b)
TO SOLVE THIS, WE FIRST REVISE SOME IDENTITIES.
1) (a + b)² = a² + 2ab + b²
2) (a - b)² = a² - 2ab + b²
3) a² - b² = (a + b)(a - b)
4) x² + (a + b)x + ab = (x + a)(x + b)
ANSWER. NOW, WE ALL KNOW (√a)² IS NOTHING BUT a ITSELF. SO LET'S REWRITE OUR EXPRESSION.
a - b ⟶ (√a)² - (√b)²
NOW, DOES THIS REMIND YOU OF A PROPERTY OR AN ALGEBRAIC IDENTITY?
YES, a² - b² IS NOTHING BUT ⟶ (a + b)(a - b).
SO IN THIS CASE, LET'S REPLACE "a" BY "√a" AND "b" BY "√b".
NOW, WE FORM AN EQUATION USING OUR IDENTITY.
(√a)² - (√b)² = (√a + √b)(√a - √b)
THEREFORE, THE FACTORS OF (a - b) ARE (√a + √b)(√a - √b).
HERE YOU GO, AN AMAZING WAY THAT REVISES YOUR LEARNING ON FACTORIZATION.
LET'S SOLVE MORE.
Q2) FIND THE FACTORS OF ⟶ (a + b)
LAST TIME WE HAD AN IDENTITY OF a² - b² = (a + b)(a - b).
BUT WE DONT HAVE ANY IDENTITY OF a + b. THEN WHAT DO WE DO? DON'T THEY HAVE ANY FACTORS? THEY OFCOURSE DO. LETS LEARN SOME ADDITIONAL IDENTITIES.
WE HAVE AN IDENTITY FOR a^3 + b^3.
WHAT IS THAT? WELL, IT IS ⟶
a^3 + b^3 = (a + b)(a² - ab + b²)
SO, NOW LET'S SOLVE THE QUESTION.
ANSWER. WE CAN WRITE a AS (∛a)^3 AND b AS (∛b)^3.
AND NOW WE CAN APPLY THE ABOVE IDENTITY OF a^3 + b^3 = (a + b)(a² - ab + b²).
SO, REPLACING "a" WITH "∛a" AND "b" WITH "∛b", WE GET THE EQUATION LIKE THIS ⟶
(∛a)^3 + (∛b)^3 = (∛a + ∛b){(∛a)² - ∛a∛b + (∛b)²}
SIMPLIFYING FURTHER, WE GET ⟶
(∛a)^3 + (∛b)^3 = (∛a + ∛b){a^2/3 - ∛(ab) + b^2/3}
THUS, THE FACTORS OF (a + b) ARE (∛a + ∛b){a^2/3 - ∛(ab) + b^2/3}
IT SEEMS LIKE a + b WAS EASIER TO LOOK AND UNDERSTAND, BUT WHEN WE FIND THE FACTORS, THEY START LOOKING A LITTLE COMPLICATED.
LET'S LOOK AT OUR LAST QUESTION OF OUR TODAY'S FACTORIZATION SESSION.
Q3) FIND THE FACTORS FOR ⟶ x² + 6x + 2.
ANSWER. THIS IS QUIET EASIER WHEN COMPARED FROM THE ABOVE QUESTIONS.
WE WILL SOLVE THIS QUESTION BY COMPLETING THE PERFECT SQUARE METHOD.
WHAT IS THIS METHOD? IT IS A METHOD USED IN FACTORISATION WHERE YOU MAKE THE TERMS A PERFECT SQUARE AND FIND THE FACTORS DEALING WITH THE THINGS THAT YOU'VE MODIFIED.
OKAY, SIMPLY PUT, LETS SOLVE THIS WHILE UNDERSTANDING.
OUR EXPRESSION x² + 6x + 2 CAN ALSO BE WRITTEN AS x² + 6x + 2 + 7 - 7. WHY DID WE ADD AND SUBTRACT SEVEN? I WILL TELL YOU IN A MOMENT.
SO NOW DO YOU SEE SOMTHING SIMILAR TO AN IDENTITY IN OUR EXPRESSION ⟶ x² + 6x + 2 + 7 - 7?
WE CAN SIMPLIFY IT AND WRITE ⟶
x² + 6x + 9 - 7
ISN'T IT?
THEN JUST LOOK AT THE FIRST 3 TERMS. DO THEY MAKE A PERFECT SQUARE? YES THEY DO. "6x" IN OUR EXPRESSION CAN ALSO BE PRIME FACTORIZED AND WRITTEN AS ⟶ 2 x 3 x (x).
THEN AS (a + b)² = a² + 2ab + b²
THEN REPLACING "a" by "x" and "b" by "9", THEN OUR EXPRESSION WILL BE : {(x)² + 2 x 3 x (x) + (3)²} - 7. AND THIS CAN BE FURTHER WRITTEN AS :
(x + 3)² - 7.
SO NOW DO YOU UNDERSTAND WHY DID WE ADD AND SUBTRACT 7 TO OUR EXPRESSION? ONLY TO MAKE THE FIRST 3 TERMS A PERFECT SQUARE. THATS THE METHOD OF COMPLETING THE SQUARE.
NOW, THE FIRST QUESTION THAT WE SOLVED WAS OUT OF THE BOX WE THINK WHEN WE ARE FACTORIZING, RIGHT? WE ASSUMED "a" AS "(√a)²" AND SO ON. SO, LETS USE OUR BRAIN HERE AGAIN.
AS YOU CAN SEE: IN (x + 3)² - 7, (x + 3)² IS WRITTEN AS A SQUARE, IF BY ANY METHOD, 7 COULD ALSO BE WRITTEN LIKE THIS, THEN CAN WE APPLY AN IDENTITY? YES!
THAT IS : a² - b² = (a + b)(a - b).
WE CAN WRITE "7" AS "(√7)²" AS WELL!
THEN OUR EXPRESSION LOOKS LIKE THIS -
(x + 3)² - (√7)².
AND NOW IT IS WAY MORE EASIER TO SOLVE OR FACTORIZE THIS EXPRESSION. USE THE IDENTITY a² - b² = (a + b)(a - b) TO GET THE FINAL FACTORS!
THUS, THE FACTORS WILL BE WRITTEN AS-
(x + 3 + √7)(x + 3 - √7)
AND THERE YOU GO, YOU'VE FOUND YOUR SOLUTION TO YOUR QUESTION!
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SO THESE WERE SOME 3 QUESTIONS ON FACTORIZATION THAT WERE AMAZING TO SOLVE, BUT THESE QUESTIONS REQUIRE OUT OF THE BOX THINKING, AND YOU KNOW WHAT IT IS.
MORE COMING SOON. BYE!
CREATED BY: NAMAN DWIVEDI
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