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X
- WE WANT TO SIMPLIFY EACH EXPRESSION
AND WRITE THE TERMS IN DESCENDING ORDER.
WE FIRST HAVE THE QUANTITY NEGATIVE X PLUS 4,
PLUS THE QUANTITY NEGATIVE 3X PLUS 7.
WHEN PERFORMING ADDITION AND SUBTRACTION OF POLYNOMIALS,
I THINK IT'S HELPFUL TO THINK OF CLEARING THE PARENTHESES
AND COMBINING LIKE TERMS.
SO, TO CLEAR THESE PARENTHESES,
WE CAN THINK OF DISTRIBUTING A POSITIVE 1.
OF COURSE, MULTIPLYING BY POSITIVE 1
IS NOT GOING TO CHANGE ANYTHING,
BUT BY DISTRIBUTING IT, WOULD CLEAR THE PARENTHESES.
SO, IF WE DISTRIBUTE A POSITIVE 1 HERE,
WE WOULD HAVE NEGATIVE X OR NEGATIVE 1X PLUS 4
AND BECAUSE WE HAVE ADDITION,
WE'RE DISTRIBUTING A POSITIVE 1 HERE AS WELL.
SO, WE'D HAVE PLUS NEGATIVE 3X OR JUST MINUS 3X PLUS 7.
AND NOW, WE'LL COMBINE LIKE TERMS.
BECAUSE WE WANT THE TERMS IN DESCENDING ORDER,
WE WANT THE X TERM FIRST, FOLLOWED BY THE CONSTANT.
SO, NOTICE HOW WE HAVE TWO X TERMS, -1X MINUS 3X.
SINCE NEGATIVE 1 MINUS 3 IS NEGATIVE 4,
THIS WOULD BE NEGATIVE 4X
AND THEN WE HAVE 4 PLUS 7, WHICH IS 11, GIVING US PLUS 11.
NEXT, WE HAVE THE QUANTITY 6X MINUS 4,
MINUS THE QUANTITY NEGATIVE 2X PLUS 5.
SO AGAIN, TO CLEAR THE PARENTHESES,
WE'LL PUT A 1 HERE AND PUT A 1 HERE,
BUT BECAUSE OF THE SUBTRACTION SIGN HERE,
WE WANT TO THINK OF DISTRIBUTING A NEGATIVE 1
TO MAKE SURE WE SUBTRACT BOTH OF THESE TERMS.
SO, DISTRIBUTING POSITIVE 1 HERE WOULD JUST GIVE US 6X MINUS 4.
AGAIN, BECAUSE OF THE SUBTRACTION SIGN HERE,
WE'LL BE DISTRIBUTING A NEGATIVE 1.
NEGATIVE 1 TIMES NEGATIVE 2X IS POSITIVE 2X, OR PLUS 2X
AND THEN, NEGATIVE 1 TIMES POSITIVE 5
IS NEGATIVE 5 OR MINUS 5.
AND NOW, COMBINING LIKE TERMS, WE HAVE 2X TERMS
6X PLUS 2X IS EQUAL TO 8X
AND NEGATIVE 4 MINUS 5 IS EQUAL TO NEGATIVE 9 OR -9
NEXT, WE HAVE NEGATIVE 7 TIMES THE QUANTITY 2X MINUS 3, -2X.
SO, WE'LL CLEAR THE PARENTHESES BY DISTRIBUTING NEGATIVE 7.
NEGATIVE 7 TIMES 2X IS NEGATIVE 14X.
NEGATIVE 7 TIMES NEGATIVE 3 IS POSITIVE 21 OR PLUS 21
AND THEN WE STILL HAVE MINUS 2X.
WE HAVE TWO X TERMS.
NEGATIVE 14X MINUS 2X WOULD BE NEGATIVE 16X
AND WE STILL HAVE PLUS 21.
AND THE LAST EXAMPLE,
WE HAVE 7X TIMES THE QUANTITY NEGATIVE 3X PLUS 5,
PLUS THE QUANTITY 3X SQUARED PLUS 4X, MINUS 8.
SO, WE'LL DISTRIBUTE POSITIVE 7X HERE
AND WE'LL PUT A 1 HERE AND DISTRIBUTE POSITIVE 1.
SO, 7X TIMES NEGATIVE 3X IS NEGATIVE 21X SQUARED
AND THEN, 7X TIMES POSITIVE 5, THAT'D BE PLUS 35X
AND THEN BY DISTRIBUTING POSITIVE 1,
IT'S NOT GOING TO CHANGE ANY OF THESE TERMS,
SO WE'LL HAVE PLUS 3X SQUARED, PLUS 4X, MINUS 8.
SO, NOW STARTING WITH THE X SQUARED TERMS,
WE HAVE NEGATIVE 21X SQUARED PLUS 3X SQUARED.
THAT'LL BE NEGATIVE 18X SQUARED
AND WE HAVE TWO X TERMS, 35X PLUS 4X, THAT'S PLUS 39X.
AND WE ONLY HAVE ONE CONSTANT TERM, MINUS 8.
AGAIN, THIS IS IN DESCENDING ORDER
BECAUSE WE HAVE THE X SQUARED TERM FIRST,
FOLLOWED BY THE X TERM, FOLLOWED BY THE CONSTANT.
I HOPE YOU FOUND THIS HELPFUL.